^wi 


THE  SLATED  ARITHMETIO. 

Entered  accorjio?  to  Act  of  Confess  in  the  year  Is6«,  by  A.  S.  BARNES  &  CO.,  in  the  Clerk'*  OfBco 
of  the  [iistrict  Court  of  the  United  St:ite«  for  the  Southern  District  of  New  York. 


SILICATE  BOOK  SLATE  SURFACE,  P«t«nt«,i  February  24, 1857;  January  15,  issr,  and  AnguU 

3ith,  1868. 


UNIVERSITY  ARITHMETIC,. 


EMBIIACING   THE 


SCIENCE  OF  NUMBERS, 


GENERAL  RULES  FOR  THEIR  APPLICATION. 


By  CHARLES  DAVIES,  LL.D., 

AUTIIOB  OF  PRIMARY,  INTELLKCTPAL  AND  SCHOOL  ARITnMKTICS;    ELEMENTARY  ALOEBEA  ; 
KLKMENTAKY   GEOMKTP.V;   PItAOTIOAL    MATHEMATICS;   ELEMENTS  OF  8UKVEVING  ; 
ELEMENTS  OF  ANALYTICAL  GEOMETRY;   nESCRIPTlVK  GEOMETRY;   SHADES, 
SHADOWS  AND    PEP^PECTIVE;   DIFFERENTIAL  AND  INTEGRAL  CAL- 
CULUS,  AND   LOGIC  AND  UTILITY  OF  MATHEMATICS. 


NEW  YORK : 

A.  S.  BARNES  &  Co.,  Ill  &  113  WILLIAM  STREET. 

BOSTON: 

WOOLWOIITH,  AINSWORTH  &  Co. 

1868. 


ADVERTISEMENT 


Th2  attention  of ■  T^acliers  U  respectfully  invitefl  to  the  Revised 

EWTIONS  of 


MiMn'  griilmeHtal  §tm% 


FOR  SCnOOLS  AND  ACADEMIES. 


1.  DAYIES'  PRIMARY  ARITHMETIC. 

2.  DAVIES'  INTELLECTUAL  ARITHMETIC. 

3.  DAVIES'  PRACTICAL  ARITHMETIC. 

4.  DAVIES'  UNIVERSITY  ARITHMETIC. 

5.  DAVIES'  PRACTICAL  MATHEMATICS. 


The  above  Works,  by  Charles  Davies,  LL.D.,  Author  of  a 
Complete  Course  of  Mathematics,  are  designed  as  a  full  Course  of 
Arithmetical  Instruction  necessary  for  the  practical  duties  of  busi- 
ness life;  and  also  to  prepare  the  Student  for  the  more  advanced 
Series  of  Matliematics  by  the  same  Author. 

The  following  New  Editions  of  Algebra^  by  Professor  Davies,  are 
commended  to  the  attention  of  Teachers: 

1.  DAVIES'  NEW  ELEMENTARY  ALGEBRA  AND  KEY. 

2.  DAVIES'  UNIVERSITY  ALGEBRA  AND  KEY. 

3.  DAVIES'  BOURDON'S  ALGEBRA  AND  KEY. 


Eatorcd  according  to  Act  of  Congress,  in  the  year  ono  thousand  eight  hundred  and 

sixty-four, 

BY    CHARLES    DAVIES, 

In   Uio   Clerk's  Office  of  the  District   Court  of  the  United   States  for 'the  Southern 
District  of  New  York. 


TO 


THE  TEACHERS  OF  THE  UNITED   STATES, 


TREATISE  ON  ARITHMETIC, 

TIIB   LAST    OF  A  SERIES    OF  "WORKS    DESIGNED   TO   LESSEN  THE    LABOB 
AND  IMPROVE  THE  SYSTEMS  OF  TEACHING, 


RESPECTFULLY    DEDICATED, 

BY 

THE  AUTHOR. 

IT  IS  OFFERED   AS   A   TOKEN   OF   HIS    GRATEFUL   APPRECIATION    OF  THE   INDULOENOB 

•WITH  wniCn  IlIB  OTTlIEn  WOBKS  HAVE  BEEIT  BlOBIVTa),  ASOt  AS  A  TESTIMONY 

OF  HIS  REGARD  FOR  THOSE  WITII  WHOM  HE  HAS  LONG  BEEN  A  00- 

UUBOUJER   IN  THE  WORK   OF  PUBLIC  INSTBUOTIOK 


PREFACE 


Science,  in  its  popular  signification,  means  knowledge  re- 
duced to  order  ;  that  is,  knowledge  so  classified  and  arranged 
as  to  be  easily  remembered,  readily  referred  to,  and  advan- 
tageously applied.  More  strictly,  it  is  a  knowledge  of  laws, 
relations,  and  principles. 

Arithmetic  is  the  science  of  numbers,  and  the  art  of  applying 
numbers  to  all  practical  purposes.  It  is  the  foundation  of  the 
exact  and  mixed  sciences,  and  an  accurate  knowledge  of  it  is 
an  important  element  either  of  a  liberal  or  practical  education. 

It  is  the  first  subject,  in  a  well-arranged  course  of  instruo- 
tion,  to  which  the  reasoning  faculties  of  the  mind  are  applied, 
and  is  the  guide-book  of  the  mechanic  and  man  of  business. 
It  is  the  first  fountain  at  which  the  young  votary  of  knowledge 
drinks  the  pure  waters  of  intellectual  truth. 

It  has  seemed,  to  the  author,  of  the  first  importance  that 
this  subject  should  be  carefully  treated  in  our  Elementary  Text- 
books. In  the  hope  of  contributing  something  to  so  desirable 
an  end,  he  has  prepared  a  series  of  arithmetical  works,  em- 
bracing four  books,  entitled.  Primary  Arithmetic  ;  Intellectua 
Arithmetic  ;  Practical  Arithmetic  ;  and  University  Arithmetic— 
the  latter  of  which  is  the  present  volume. 

Primary  Arithmetic.  This  first-book  is  adapted  to  the 
capacities  and  wants  of  young  children.  Sensible  objects  are 
employed  to  illustrate  and  make  familiar  the  simple  combina- 
tions and  relations  of  numbers.  Each"  lesson  embraces  ono 
combination  of  numbers,  or  one  set  of  combinations. 


6  PRE}"ACE. 

Intellectual  Arithmetic.  This  work  is  designed  to  present 
a  thorough  analysis  of  the  science  of  numbers,  and  to  form  a 
complete  course  of  mental  arithmetic.  I  have  aimed  to  make 
it  accessible  to  young  pupils  by  the  simplicity  and  gradation 
of  its  methods,  and  to  adapt  it  to  the  wants  of  advanced 
students  by  a  scientific  arrangement  and  logical  connection,  in 
11  the  higher  processes  of  arithmetical  analysis. 

Practical  Arithmetic.  Great  pains  have  been  taken,  in  the 
preparation  of  this  book,  to  combine  theory  and  practice;  to 
explain  and  illustrate  j^rinciiDles,  and  to  apply  them  to  the  com- 
mon business  transactions  of  life — to  make  it  emphatically  a 
practical  work.  The  student  is  required  to  demonstrate  every 
principle  laid  down,  by  a  course  of  mental  reasoning,  before 
deducing  a  proposition  or  making  a  practical  application  of  a 
rule  to  examples.  He  is  required  to  fix  and  apprehend  the 
unit  or  base  of  all  numbers,  whether  integral  or  fractional — 
to  reason  with  constant  reference  to  this  base,  and  thus  make 
it  the  key  to  the  solution  of  all  arithmetical  questions.  It  is 
hoped,  that  the  language  used  in  the  statement  of  principles, 
in  the  definition  of  terms,  and  in  the  explanation  of  methods, 
will  be  found  to  be  clear,  exact,  brief,  and  comprehensive. 

University  Arithmetic.  This  work  is  designed  to  answer 
another  object.  Here,  the  entire  subject  is  treated  as  a  science. 
The  pupil  is  supposed  to  be  fAmillar  with  the  simple  operations 
in  the  four  ground  rules,  and  with  the  first  principles  of  frac- 
tions, these  being  now  taught  to  small  children,  either  orally  or 
from  elementary  treatises.  This  being  premised,  the  language 
of  figures,  which  are  the  representatives  of  numbers,  is  care- 
fully taught,  and  the  different  significations  of  which  the  figures 
themselves  are  susceptible,  depending  on  the  manner  in  which 
they  are  written,  are  fully  explained.  It  is  shown,  for  example, 
that  the  simple  numbers  in  which  the  value  of  the  unit  increases 
from  right  to  left  according  to  the  scale  of  tens,  and  the  De- 
nominate or  Compound  numbers  in  which  it  increases  according 
to  a  varying  scale,  belong  to  the  same  class  of  numbers,  and 


that  both  may  be  treated  under  tke  .same  rules,  Heace,  the 
rules  for  Notation,  Addition,  Subtraction,  Multiplication,  and 
Division,  have  been  so  constructed  as  to  apply  equally  to  all 
numbers.  This  arrangement,  which  the  author  has  not  seen 
elsewhere,  is  deemed  an  essential  improvement  in  the  science  of 
^I'ithmctic.  ;^  .. 

In  developing  the  properties  of  mmaberSj  from  their  elemcn 
xnry  to  tlicir  highest  combinations,  great  labor  has  been  be- 
stowed on  classification  and  arrangement.  It  has  been  a  lead- 
ing object  to  present  the  entire  subject  of  arithmetic  as  form- 
ing a  series  of  dependent  and  coimected  projyosUions :  so  that 
the  i)upil,  while  acquiring  useful  and  practical  knowledge,  may 
at  the  same  time  be  introduced  to  those  beautiful  methods  of 
reasoning  which  science  alone  teaches. 

Groat  care  has  been  taken  to  demonstrate  every  proposition 
— to  give  a  complete  analysis  of  all  tlie  .methods  employed, 
from  the  simplest  to  the  most  difficult,  and  to  explain  fully 
tlie  reason  of  every  rule.  A  full  analysis  of  the  science  of 
Numbers  has  developed  but  one  law;  viz.,  the  law  which  con- 
vects  all  the  numbers  of  arithmetic  with  the  unit  one,  and 
which  points  out  the  relations  of  these  numbers  to  each  other. 

In  the  Appendix,  which  treats  of  Units,  Weights,  and  Meas- 
ures, &c.,  the  methods  of  determining  the  Arbitrary  Unit,  as 
well  as  the  general  law  which  prevails  in  the  formation  of 
numbers,  are  fully  explained.  I  cannot  too  earnestly  recom- 
mend this  part  of  the  work  to  the  special  attention  of  Teach- 
ers and  pupils. 

In  fine,  the  attention  of  Teachers  is  especially  invited  to  this 
work,  because  general  methods  and  general  rules  are  employed 
to  abridge  the  common  arithmetical  processes,  and  to  give  to 
them  a  more  scientific  and  practical  character.  In  the  present 
edition,  the  matter  is  presented  in  a  new  form  ;  the  arrange- 
ment of  the  subjects  is  more  natural  and  scientific ;  the  metliod;^ 
have  been  carefully  considered;  the  illustrations  abridged  and 
simplified  ;  the  definitions  and  rules  thoroughly  revised  and  cor- 


8  rPwEFACE. 

rected  ;  and  a  very  large  number  and  variety  of  practical  ex- 
amples have  been  added.  The  subjects  of  Fractions,  Propor- 
tion, Interest,  Percentage,  Alligation,  Analysis,  and  Weights 
and  Measures,  present  many  new  and  valuable  features,  which 
are  not  found  in  other  works. 

A  Key  to  the  present  work  has  also  been  published  for  the 
use  of  such  Teachers  as  may  desire  it, — prepared  with  great 
care,  containing  not  only  the  answers  and  solutions  of  all  the 
examples,  but  a  full  and  comprehensive  analysis  of  the  more 
difficult  ones. 

The  author  has  great  pleasure  in  acknowledging  the  interest 
which  Teachers  have  manifested  in  the  success  of  his  labors  : 
they  have  suggested  many  improvements,  both  in  rules  and 
methods,  not  only  in  his  elementary,  but  also  in  his  advanced 
works.  The  recitation-room  is  the  final  tribunal,  and  the  intel- 
ligent teacher  the  fiiml  judge,  before  which  all  text-books  must 
stand  or  fall. 

CoLUJNrBiA  College,) 
May,  1864.         J 


CONTENTS-. 


FIRST    FIVE    RULES. 

PAQX 

Definitions 13 

Ex-pressing  Numbers 14 

Notation  and  Numeration 14-24 

Formation  and  Nature  of  Numbers 24 

Scales 25-28 

Integral  Units 28 

Properties  of  the  9's 29 

Eduction 80-34 

Addition 34-44 

Subtraction 44-54 

Multiplication 54-C8 

Division 68-87 

Practice 87-90 

Longitude  and  Time 00-94 

Applications  in  tlie  Four  Rules 9 1-103 

Properties  of  Numbers 103 

Divisibility  of  Numbers 103-105 

Cancellation lOG-109 

Least  Common  Multiple 109-111 

Greatest  Common   Divisor 111-115 


COMMON    FRACTIONS. 

Definition  of,  and  First  Principles 115-119 

The  six  Kinds  of  Fractions 119-120 

Six  Propositions 120-124 

llwduction  of  Common  Fractions 124-133 

Reduction  of  Denominate  Fractions 132-135 

Addition  of  Common  Fractions 135-140 

Subtraction  n\'  CnirTSKm  Frp/^tions 140-145 


10  CONTENTS. 

PACK 

Multiplication  of  Common  Fractions 145-148 

Division  of  Common  Fractions 148-152 

Complex  Fractions 153 

Applications  in  Fractions 153-155 


DUODECIMALS. 

Principles  and  Rules 155-1G4 

DECIMAL    FRACTIONS. 

Definition  of  Decimals,  &c.. 1G4 

Decimal  Numeration  Table — First  Principles,  &c 1G5-170 

Addition  of  Decimals.. 170-173 

Subtraction  of  Decimals , .  173-175 

Multiplication  of  Decimals 175-177 

Contractions  in  Multiplication 177-179 

Division  of  Decimals 179-184 

Contractions  in  Division 184-186 

Reduction  of  Common  Fractions  to  Decimals 180-190 

Reduction  of  Denominate  Decimals 190-191 

Repeating  Decimals — DejSnition  of,  &c 191-195 

Reduction  of  Repeating  Decimals 195-201 

Addition  of  Repeating  Decimals 201 

Subtraction  of  Repeating  Decimals 201 

Multiplication  of  Repeating  Decimals 202 

Division  of  Repeating  Decimals 203 

CONTINUED  FEACTIONS. 

Definitions  and  Principles ' 203-206 


RATIO  AND  PROPORTION. 

Ratio  Defined 206 

Compound  Ratio 207 

Simx)le  Proportion 209 

Direct  and  Inverse  Proportion , 211 

Single  Rule  of  Three 214-220 

Double  Rule  of  Three 231-235 

Partnership 225  -2n0 


CONTENTS.  l^t 

PEBCENTAGE. 

•  PAQK 

Percentage  Defined  and  Ill-ustrated  ...iitj*tii¥fitfti:,..i...  331-230 

Profit  and  Lobs ..^.v.^^^^....... ...  237-241 

Commission 242  244 

Interest ^^»  ^. ,  ^ 245-253 

Partial  Payments ,.., . ,^. .^. . . . . .,. . . .^^,^. . .  253-25G 

Problems  in  Interest.. :...;..::;:?.  .■Tr'^'l'^.^^T:"'^'..';:.  ..^        250-257 

Compound  Interest 258-259 

Discount \. . , . i .'i . i; H; 200-201 

Banking ...,.^»^.,.,.. ..."....... ...  202-204 

Bank  Discount « tu.. 204-206 

Stocks ....^«...... 207-272 

Insurance ..««.»».,•- 273-275 

Liie  Insurance It^v:  V.'.^^ 275-277 

Endowments 278 

Annuities 279-281 


AI'FLICATIOKS. 

Assessing  Taxes .-...".. 281-284 

Equation  of  Payments .'.v.v.'.v.-. 284-294 

Alligation ..'............ 295 

Alligation  Medial. 295 

Alligation  Alternate. .'. 290-301 

Custom-house  Business 202-200 

Tonnage  of  Vessels .' ". 300-307 

General  Average '."..'. 308-310 

Coins  and  Currencies. .. .'.'.V .".'/. v. 311 

Exchange ...............  ;<>.¥.  Im  .*Vw^ 312-319 

Arbitration  of  Exchange. ...... ...;;::.-;;.-t;;8i;^. 820-332 

POWERS  AND  ROOTS. 

Involution 322 

Evolution 323 

Extraction  of  Square  Root 324-333 

Cube  Root 333-339 

ARirnMETICAL  PROGRESSION. 

Definition  of,  &c 330 

Different  Cases a40-3^4 


12  CONTENTS. 

GEOMETRICAL   PROGRESSION. 

Different  Cases 345-348 


PAGK 

Definition  of,  &c... : 344 


ANALYSIS. 

Analysis  and  Promiscuous  Examples 348-869 

MENSURATION. 

Mensuration  of  Surfaces. 370-375 

Mensuration  of  Volumes 375-380 

Gauging 381-383 

Mechanical  Powers 384-393 

Questions  in  Natural  Pliilosophy 392-398 

APPENDIX. 

Different  Kinds  of  Units 399-403 

Abstract  Units 403 

Units  of  Currency 404-408 

Linear  Units 408-409 

Units  of  Surface 410 

Units  of  Volume 411-413 

Units  of  Weight 414-41G 

Units  of  Time 416-418 

Units  of  Circular  Measure 419 

Miscellaneous  Table 419 

Books  and  Paper , 420 

Metric  System  of  Weights  and  Measures 421-432 

Answers 433-466 


UNIVERSITY  ARITHMETIC. 


Definitions. 

1.  A  Unit  is  a  single  tiling,  or  one. 

2.  A  Number  is  a  unit,  or  a  collection  of  units. 

3.  Science  treats  of  the  properties  and  relations  of  things  : 
Art  is  the  practical  application  of  the  principles  of  Science. 

4.  Arithmetic  is  the  Science  of  Numbers,  and  also  the  Art 
of  applying  numbers  to  practical  purposes. 

5.  A  Proposition  is  something  to  be  done,  or  demonstrated. 

6.  An  Analysis  is  an  examination  of  the  separate  parts 
of  a  proposition. 

7.  An  Operation  is  the  doing  of  something  with  numbers. 

8.  A  Rule  is  the  direction  for  performing  an  operation. 

9.  An  Answer  is  the  result  of  a  correct  operation. 

Operations  of  Arithmetic. 

10.  There  are,  in  Arithmetic,  five  fundamental  operations : 
Notation  and  Numeration,  Addition,  Substraction,  Multiplica- 
tion, and  Division. 


1.  What  is  a  Unit? — 3.  What  is  a  Number? — 3.  Of  what  does  Scionco 
treat?  What  is  Art? — 4.  What  is  Arithmetic?—^.  What  is  a  Pn^posi- 
tion?— G.  What  is  an  Analysis? — 7.  What  is  an  Operation ?— 8.  AVlmt 
is  a  Rule? — 9.  What  is  an  Answer? — 10.  IIow  many  fundamental  oper- 
ations are  there  in  Arithmetic?    What  are  thoy? 


14 


NOTATION   AND  NUMERATION. 


Expressing  Numbers. 
11.    There  are  three  methods  of  expressing  numbers: 
1.    By  words,  or  common  language ; 
;  ,2.    By  letters,  called  the  Roman  method ; 
'3.    By  figures,  called  the  Arabic  method. 


Expressing  Numbers  by  "Words. 


12.  A  single  thing  is  called 
One  and  one  more 
Two  and  one  more 
Three  and  one  more 
Four  and  one  more 
Five  and  one  more 
Six  and  one  more 
Seven  and  one  more 
Eight  and  one  more 
Nine    and  one  more 


Each  of  the  words,  one,  two,  thr 
number,  and  denotes  how  many  units  are  taken 
are   generally  called  numbers  ;    though,  in  fac 
the  names  of  numbers. 


One, 

Two. 

TJiree. 

Four. 

Five. 

Six. 

Seven. 

Eight. 

Nine. 

Ten. 

ee,  four,  &c.,  expresses  a 

These  words 

they   are   but 


NOTATION  AND   NUMERATION. 

13.  Notation  is  the  method  of  expressing  numbers,  either 
by  letters  or  figures. 

Numeration  is  the  art  of  reading,  correctly,  any  number 
expressed  by  letters  or  figures. 

There  are  two  methods  of  Notation  :  the  one  by  letters,  the 
other  by  figures.  The  method  by  letters  is  called  the  Roman 
Notation;  the  method  by  figures  is  called  the  Arabic  Notation, 


NOTATION   AND   NUMEEATION. 


15 


Roman  Notation. 
14.    In  the  Roman  Notation,  seven  capital  letters  are  used. 
They  express  the  following  values  : 


I 


Y 

five, 


X 

ter 


L 

fifty, 


c 


one 
hundred, 


D 

five 
hundred. 


M 


All  other  numbers  arc  expressed  by  combining  these  letters, 
according  to  the  following  principles : 

1.  Every  time  a  letter  is  repeated,  the  number  which  it  de- 
notes is  repeated. 

2.  If  a  letter  denoting  a  less  number  be  written  on  the  right 
of  one  denoting  a  greater^  the  number  expressed  will  be  the 
stem  of  the  numbers. 

3.  If  a  letter  denoting  a  less  number  be  written  on  the  left 
of  one  denotmg  a  greater,  the  number  expressed  will  be  the 
difference  of  the  numbers. 

4.  A  dash  ( — ),  placed  over  a  letter,  increases  the  number 
for  which  it  stands,  a  thousand  times. 


I 

II 

III 

IV 

V 

VI 

VII 

VIII 

IX 

X 

XX 

XXX 

XL 

L 

LX 

LXX 


Roman  Table 

. 

One. 

LXXX 

.    Eiglity. 

Two. 

xc 

.    Ninety. 

Three. 

c 

.     One  hundred. 

Four. 

cc 

.     Two  hundred. 

Five. 

ccc 

.    Three  hundred. 

Six. 

cccc 

.    Four  hundred. 

Seven. 

D 

.    Five  liundred. 

Eight. 

DC 

.     Six  hundred. 

Nine. 

DCC 

.    Seven  hundred. 

Ten. 

DCCC 

.    Eight  hundred. 

Twenty. 

DCCCC 

J        .    Nine  hundred. 

Thirty. 

M 

.    One  thousand. 

Forty. 

MD 

.    Fifteen  hundred. 

Fifty. 

MM 

.    Two  thousand. 

Sixty. 

V 

Five  thousand. 

Seventy. 

X 

.    Ten  thousand. 

ion  was  used 

by  the  Ro 

mans    hence  its  name. 

numbering 

chnpteip, 

pages,  &c. 

It 


Id 


NOTAI.ON   AND  NUMEEATION. 


Ezamples 

Express  tlie  following  numbers  in  Roman  Notation : 

1. 

Eleven. 

23. 

Eighty-one. 

2. 

Fourteen. 

24. 

Eighty-seven. 

3. 

Sixteen. 

25. 

Eighty-nine. 

4. 

Seventeen. 

26. 

Ninety-four. 

5. 

Nineteen. 

2t. 

Ninety-five. 

6. 

Twenty-two. 

28. 

Ninety-seven. 

t. 

Twcnty-eiglit 

29. 

Ninety-nine. 

8. 

Twenty-nine. 

30. 

One  hundred  and  fifteen. 

9. 

Thirty-three. 

31. 

Seven  hundred  and  fifty. 

10. 

Thirty-seven. 

32. 

One  thousand  and  sixty. 

11. 

Thirty-eight. 

33. 

Two  thousand  and  forty. 

12. 

Forty-three. 

34. 

Five  hundred  and  sixty. 

13. 

Forty-seven. 

35. 

Nine  hundred  and  sixty, 

14. 

Forty-nine. 

36. 

Six  hundred  and  ninety. 

15. 

Fifty-six. 

3t. 

One  thousand  and  fifty. 

16. 

Fifty-eight. 

38. 

Four  thousand  and  four. 

n. 

Fifty-nine. 

39. 

Six  thousand  and  nine. 

18. 

Sixty-five. 

40. 

Nine  thousand  and  nine. 

19. 

Sixty-nine. 

41. 

Eight  hundred  and  six. 

20. 

Sixty-seven. 

42. 

Six  hundred  and  eight. 

2L 

Seventy-five. 

43. 

Eight  thousand  and  six. 

22. 

Seventy-six. 

44. 

Two  thousand  and  one. 

11.  How  many  methods  are  there  of  expressing  numbers?  Wliat  aro 
they  ? 

12.  What  does  each  of  the  words,  one,  two,  three,  &c.,  denote? 
What  are  these  words  generally  called?      What  are  they,  in  fact? 

13.  What  is  Notation?    What  is  Numeration?    How  many  methods 
f  Notation  are  there  ?    What  are  they  ? 

14.  How  many  letters  does  the  Roman  notation  employ  ?  "SVhich  are 
they?  What  value  does  each  represent?  What  is  the  efiect  of  repeat- 
ing a  letter  ?  What  is  the  number,  when  a  letter  denoting  a  less  number 
is  placed  on  the  right  of  one  denoting  a  greater?  AYhat  is  the  number, 
when  a  letter  denoting  a  less  number  is  placed  on  the  left  of  one  de- 
noting a  greater?    What  is  the  efi(3Ct.  of  placing  a  dash  over  a  letter? 


NOTATION  AND  NUMERATION.  IT 

Arabic  Notation. 

15.  Arabic  Notation  is  the  method  of  expressing  numbers 
by  figures.  Ten  figures  are  used,  and  they  form  the  Alphabet 
of  the  Arabic  Notation.     They  are, 

01234         56^89 

naught,       one,         two,        three,        four,        five,  six,        seven,       eight,       nine. 

The  naught,  0,  is  also  called  cipher.  It  denotes  no  number 
)ut  the  absence  of  a  thing.  Thus,  if  there  are  no  apples  in  a 
basket,  we  write,  the  number  of  apples  in  the  basket  is  0 
The  other  nine  figures  are  called  Significant  Figures,  or  Digits 

Orders  of  Units. 

16.  We  have  no  single  figure  for  the  number  ten.  We 
therefore  combine  the  figures  already  known.  This  we  do  by 
writing  0  on  the  right  hand  of  1: 

Thus, 10 

which  is  read,  ten. 

This  10  is  equal  to  ten  of  the  units  expressed  by  1.  It  is, 
however,  but  a  single  ten,  and  may  be  regarded  as  a  unit,  ten 
times  as  great  as  the  unit  1.  It  is  called,  a  unit  of  the  second 
order. 

17.  When  two  figures  are  written  by  the  side  of  each  other, 
the  one  on  the  right  is  in  the  place  of  units,  and  the  other  in 
the  place  of  tens,  or  of  units  of  the  second  orde".  Each  unit 
of  the  second  order  is  equal  to  ten  units  of  the  first  order. 

When  units  simply  are  named,  units  of  the  first  order  are 
always  meant. 

Units  of  the  second  order  are  written  thus  : 


One  ten,  or  ...     . 

.    10 

Six  tens,  or  sixty, 

.    60 

Two  tens,  or  twenty, 

.    20 

Seven  tens,  or  seventy, 

.    70 

Threti  tons,  or  thirty, 

.    30 

Eight  tens,  or  eighty. 

.    80 

Four  tens,  or  forty,   . 

.    40 

Nine  tens,  or  ninety. 

.    90 

Rve  tens,  or  fifty,     . 

.    50 

One  hundred,    .     .    . 

.  100 

1-8  KOTATIOl^   AND  >TtrMERATION. 

18.   To  express  ten  imits  of  the  second  order,  or  one  hundred^ 
we  form  a  new  combination  : 

Thus,     . 100 

by  writing  two  ciphers  on  the  right  of  1.     This  number  is  read, 
one  hundred,  and  is  a  unit  of  the  third  order. 

We  can  now  express  any  number  less  than  one  thousand. 

In  the  number  two  hundred  and  fifty-five,  there 

are  5  units,  5  tens,  and  3  hundreds.     Write,  there-  g    «;    .^ 

fore,  5  units  of  the  first  order,  5  units  of  the  second  ^    ^    | 

order,  and  2  of  the  third ;  and  read  from  the  right,  2    5    5 

units,  tens,  hundreds;  and  from  the  left,  two  hundred 
and  fft7j-five. 

In  the  number  five  hundred  and  ninety-five,  there  g    »  ^ 

are  5  xmits  of  the  first  order,  9  of  the  second,  and  ,§-2  3 

five  of  the  third ;  and  it  is  read  from  the  right,  units,  5    9  5 
tens,  hundreds. 


In  the  number  six  hundred  and  four,  there  are  §    g   .^ 

4  miits  of  the  first  order,  0  of  the  second,  and  6  of  ^    ^    3 

the  third.  6    0    4 


The  right-hand  figure  always  exjjresses  units  of  the  first 
order;  the  second,  units  of  the  second  order ;  and  the  third, 
units  of  the  third  order, 

19.  To  express  ten  units  of  the  third  order,  or  one  thousand^ 
we  form  a  new  combination  : 

Thus, 1000 

by  writing  three  ciphers  on  the  right  of  1.     This  num])er  is 
read,  one  thousand,  and  is  a  unit  of  Wiq  fourth  order. 

We  may.  now  form  as  many  orders  of  units  as  we  please  : 

A  single  unit  of  the  first  order  is  expressed  by  ...  .  1 

A  unit  of  the  second  order  by  1  and  0;  thus,  .        .        .  .        10 

A  unit  of  the  third  order  by  1  and  two  O's ;  .        .         .  .      100 

A  unit  of  the  fourth  order  by  1  and  three  O's ;  .        ,        .  .     1000 

A  imit  of  the  fifth  order  by  1  and  four  O's ;  .        .        .  .  10000 

And  so  on,  for  units  of  higher  orders. 


NOTATION   AND  NUMERATION.  lU 

Hence,  tlie  following  principles : 

1st.  The  same  figure  expresses  different  units  according  to 
the  place  which  it  occupies: 

2d.  Units  of  the  first  order  occiqyy  the  place  at  the  right; 
units  of  the  second  order,  the  second  place ;  units  of  the  third 
order,  the  third  place;  and  the  unit  of  any  figure  is  deter 
mined  hy  the  number  of  its  place: 

3d.  Ten  units  of  the  first  order  make  one  of  the  second ^ 
ten  of  the  second,  one  of  the  third;  ten  of  thd  third,  one  oj 
the  fourth ;   and  so  on  for  the  higher  orders: 

4th.  When  figures  are  written  hy  the  side  of  each  other, 
ten  units  in  any  one  place  maJce  one  unit  of  the  place  next 
at  the  left. 

Examples  in  Writing  the  Orders  of  Units. 

1.  Write  7  units  of  the  1st  order. 

2.  Write  8  units  of  the  2d  order. 

3.  Write  9  units  of  the  4  th  order. 

4.  Write  3  units  of  the  1st  order,  with  9  of  the  2d. 

15.  What  is  the  Arabic  Notation?  How  many  figures  are  used? 
What  do  tliey  form?  Name  the  figures.  What  does  0  express? 
WTiat  are  the  other  figures  called? 

IG.  Have  we  a  separate  character  for  ten?  How  do  we  express  ten? 
To  how  many  units  1  is  1  ten  equal  ?  May  ten  bo  regarded  as  a  single 
unit  ?    Of  what  order  ? 

17.  When  two  figures  are  written  by  the  side  of  each  other,  wliat 
place  docs  the  right-band  figure  occupy?  The  figure  on  the  left? 
When  units  simply  arc  named,  wliat  units  are  meant? 

18.  How  do  you  write  one  hundred?  To  how  many  units  of  the 
second  order  is  it  equal  ?  To  how  many  of  the  first  order  ?  How  may 
it  be  regarded  ?  Of  what  order  ?  How  many  units  of  the  third  order 
in  200?     In  GOO?     In  900? 

19.  To  what  arc  ten  units  of  the  third  order  equal?  How  do  you 
write  it  ?  How  do  you  write  a  single  unit  of  the  first  order  ?  How  do 
you  write  a  unit  of  the  second  order?  Of  the  third?  Of  the  fourth? 
Ten  units  of  the  first  order,  make  what  ?  Ten  of.  any  order,  make  what  ? 
When  figures  are  written  by  the  side  of  each  other,  how  many  unita 
of  any  place  make  one  unit  of  the  place  next  to  the  left  ? 


20  NOTATION  AND  NUMEEATION. 

6.  Write  9  units  of  the  3d  order,  with  6  of  the  2d,  and  1 
of  the  1st. 

6.  Write  0  units  of  the  2d  order,  8  of  the  1st,  with  4  of 
the  3d,  and  1  of  the  4th. 

I.  Write  8  units  of  the  6th  order,  t  of  the  4th,  9  of  the 
5th,  0  of  the  3d,  2  of  the  2d,  and  1  of  the  1st. 

8.  Write  8  units  of  the  8th  order,  6  of  the  Tth,  0  of  the 
1st,  3  of  the  2d,  4  of  the  3d,  9  of  the  4th,  0  of  the  6th,  and 
2  of  the  5th. 

9.  Write  4  units  of  the  10th  order,  8  of  the  1th,  3  of  the 
9th,  2  of  the  8th,  0  of  the  6th,  3  of  the  1st,  6  of  the  2d,  0 
of  the  3d,  1  of  the  4th,  and  2  of  the  5th. 

10.  Write  3  units  of  the  2d  order,  2  of  the  1st,  9  of  the 
3d,  0  of  the  4th,  9  of  the  9th,  6  of  the  8th,  t  of  the  tth,  0 
of  the  6th,  and  4  of  the  5th. 

II.  Write  3  units  of  the  11th  order,  0  of  the  10  th,  8  of 
the  4th,  0  of  the  5th,  2  of  the  6th,  0  of  the  Ith,  3  of  the 
•8th,  4  of  the  9th,  1  of  the  3d,  2  of  the  2d,  and  3  of  the  1st. 

12.  Write  3  units  of  the  12th  order,  6  of  the  11th,  3  of 
the  8th,  7  of  the  6th,  2  of  the  4th,  and  1  of  the  2d. 

13.  Write  5  units  of  the  13th  order,  8  of  the  12th,  0  of 
the  9th,  6  of  the  Tth,  8  of  the  3d,  and  12  of  the  1st. 

14.  Write  T  units  of  the  14th  order,  5  of  the  13th,  6  of 
the  12th,  5  of  the  10th,  1  of  the  8th,  9  of  the  6th,  5  of  the 
4  th,  and  8  of  the  1st. 

15.  Write  9  units  of  the  15th  order,  4  of  the  13th,  8  of 
the  9th,  2  of  the  6th,  1  of  the  3d,  and  2  if  the  2d. 

16.  Write  6  units  of  the  16th  order,  9  of  the  12th,  1  of 
the  9th,  4  of  the  1th,  0  of  the  6th,  8  of  the  4th,  9  of  the 
5th,  and  2  of  the  2d. 

11.  Write  8  units  of  the  20th  order, 
the  13th,  4  of  the  11th,  9  of  the  9th, 
the  5th,  and  9  of  the  3d. 

18.  Write  6  units  of  the  10  th  order, 
the  6th,  0  of  the  4th,  and  1  of  the  1st 


5  of  the  18th, 

6  of 

1  of  the  mh, 

4  of 

5  of  the  8th, 

9  of 

NOTATION    AND   NUMEEAITON. 


21 


19.   Write  9  units  of  the  ISth  order,  and  then  dhnuiish  the 
figure  of  each  order  by  1  till  you  come  to  and  mclude  0  ;  then 


the  figure  of  each  order  by  1,  till  you  reach  the  first 
order. 


increase 


order ;   and  then  read  each 


Numeration  Table 


7th  Period. 
Quintillions. 


6th  Period. 
Quadrilliona. 


6th  Period. 
Trillions. 


4th  Period. 
Billions. 


Sd  Period. 
Millions. 


2d  Period. 
Thousands 


o 

j 

Ct-I 

O 
00 

(=1 

w 


i-l 


EH         H 


W 


W 


'S 


«   2 


a  -a 
H  0 


70,804,21G,G3G,806,304,625 


Notes. — 1.  Numbers  expressed  by  more  than  three  figures  are  writ- 
ten and  read  by  periods,  as  shown  in  the  above  table. 

2.  Each  period  always  contains  three  figures,  except  the  left-hand 
period,  which  may  contain  one,  two,  or  three  figures. 

3.  Thfe  unit  of  the  first,  or  right-hand  period,  is  1;  of  the  second 
period,  1  thousand;  of  the  third,  1  million;  of  the  fourth,  1  billion; 
and  so  on,  for  periods,  stiU  to  the  left. 

4.  To  QuintiUions  succeed  Sextillions,  SeptiUions,  Octillions,  Nonil- 
lions,  Decillions,  Undecillions,  Duodecillions,  &c. 

5.  The  pupils  should  be  required  to  commit,  thoroughly,  the  names 
of  the  periods,  so  as  to  repeat  them  in  their  regular  order  from  left 
to  right,  as  well  as  from  right  to  left. 

6.  Formerly,  in  the  English  Notation,  six  places  were  given  to 
Millions.  They  were  read.  Millions,  Tens  of  Millions,  Hundreds  o 
Millions,  Thousands  of  Millions,  Tens  of  Thousands  of  MlUions,  Hundreds 
of  Thousands  of  Millions.  This  method  produced  great  irregularity  in 
the  Notation,  as  it  gave  three  places  to  the  units  of  the  first  two  periods 
(viz.;  units  and  thousands),  and  six  places  to  the  next  denomination. 
The  French  method,  which  gives  three  places  to  the  unit  of  each  period, 
is  fully  adopted  in  this  country,  and  must  soon  become  universal. 


^2  NOTATION   AND   NUMERATION. 

Notation  and  Numeration. 

Rule  for  Notation. 

I.  Begin  at  the  left  hand  and  write  each  period  in  order, 
as  if  it  were  a  period  of  units: 

TI    When  the  number,  in  any  period  except  the  left-hand 

period,  can  he  expressed  by  less  than  three  figures,  prefix  one 

or  two  ciphers;  and  ivhen  a  vacant  period  occurs,  fill  it  with 

ciphefs. 

Rule  for  Numeration. 

I.  Separate  the  number  into  periods  of  three  figures  each^ 
beginning  at  the  right  hand: 

II.  Name  the  unit  of  each  figure,  beginning  at  the  right: 

III.  Then,  beginning  at  the  left  hand,  read  each  period  as 
if  it  stood  alone,  naming  its  unit. 

Examples  for  Practice. 
Express  the  following  numbers  in  figures. 

1.  Six  hundred  and  twenty-one. 

2.  Five  thousand  seven  hundred  and  two. 

3.  Eight  thousand  and  one. 

4.  Ten  thousand  four  hundred  and  six. 

5.  Sixty-five  thousand  and  twenty-nine. 

6.  Forty  millions  two  hundred  and  forty-one. 
1.   Fifty-nine  millions  three  hundred  and  ten. 

8.  Eleven  thousand  eleven  hundred  and  eleven. 

9.  Three  hundred  millions  one  thousand  and  six. 

10.  Sixty-nine  billions  three  millions  and  two  hundred. 

Let  the  pupil  point  off  and  read  the  following  numbers  ;  then 
wi'ite  them  in  words  : 

11.  91          16.        3204560*1  21.  184236104 

12.  326  It.        90464213  22.         T403026054 

13.  3302  18.        47364291  23.       21104080495 

14.  65042  19.    4031902169  24.       21896120421 

15.  142604  20.        91046302  25.  8140290308091 


NO-mTION  AND  NUMEBATION.  23 


26.     85046804GY023 
21.  90403040t20l5G 

28.    172304136893210 


29.  30461214302704 

30.  167320410341204 

31.  2164032189765421 


Let  each  of  the  above  examples,  after  being  written  on  the 
blackboard,  be  analyzed  as  a  class  exercise  ;   thus — 

1.  In  how  many  ways  may  the  number  97  be  read? 
1st.   The  common  way,  ninety-seven. 

2d.   We'  may  read,  9  tens,  arid  *l  units. 

2.  Im  how  many  ways  may  326  be  read? 

1st.   By  the  common  way,  three  hundred  and  twenty-six. 
2d.   Three  hundred,  2  tens,  and  6  units. 
3d.   Thirty-two  tens,  and  six  units. 

3.  In  how  many  ways  may  the  number  5302  be  read? 
1st.  Five  thousand  three  hundred  and  two. 

2d.   Five  thousand,  three  hundred,  0  tens,  and  2  units. 
3d.   Fifty-three  hundi'ed,  0  tens,  and  2  units. 
4th.   Five  hundred  and  thirty  tens,  and  2  units. 

4.  In  65042,  how  many  ten  thousands?  How  many  thou- 
sands ?  How  many  hundreds  ?  How  many  tens  ?  How  many 
units  ? 

6.  In  742604,  how  many  hundred  thousands  ?  How  many 
ten  thousands  ?  How  many  thousands  ?  How  many  hundreds  ? 
How  many  tens?     How  many  units? 

Let  the  pupil  express  the  following  in  figures : 

32.  Forty-seven  quadrillions,  sixty-nine  billions,  four  hundred 
and  sixty-five  thousand,  two  hundred  and  seven. 

33.  Eight  hundred  quintillions,  four  hundred  and  twenty-nine 
millions,  six  thousand  and  nine. 

34.  Ninety-five    sextillions,    eighty-nine    millions,    eighty-m'n 
housand,  three  hundred  and  six. 

35.  Six  quintillions,  four  hundred  and  fifty-one  bilhons,  sixty 
five  millions,  forty-seven  thousand,  one  hundred  and  four. 

36.  Nme  hundred  and  ninety-nine  billions,  sixty-five  millions, 
eight  Imndred  and  forty-one  thousand,  four  hundred  and  eleven. 


24  FORMATION   OF   NUMBWiS. 

Formation  of  Numbers. 

20.  One  refers  to  any  single  tJdng,  and  has  no  reference  to 
kind  or  quality.     It  is  called  an  Abstract  Unit. 

One  foot  refers  to  a  single  foot,  and  is  called  a  Denominate 
or  Concrete  Unit. 

21.  An  Abstract  Number  is   one  whose  unit   is   abstract 
(hus,  three,  four,  six,  &c.,  are  abstract  numbers. 

22.  A  Denominate  or  Concrete  Number  is  one  whose  unit 
is  denominate  or  concrete  ;  thus,  three  feet,  four  dollars,  five 
pounds,  &c.,  are  denominate  numbers. 

23.  A  Simple  Number  is  a  single  unit,  or  a  single  collection 
of  units,  either  abstract  or  denominate. 

Two  numbers  are  of  the  same  denomination  when  they  have 
the  same  unit ;  and  of  different  denominations  when  they  have 
different  units. 

24.  A  Compound  Denominate  Number  is  one  expressed  by 
two  or  more  different  units ;   as,  1  yard  2  feet  6  inches. 

Laws  of  the  Units  and  Scales. 

25.  We  have  seen  that  when  figures  are  written  by  the  side 
of  each  other,  thus, 

6  18  9  0  4, 
the  language  implies  that  ten  units,  of  any  place,  make  one  unit 
of  the  place  next  to  the  left. 

When  figures  are  written  to  express  English  Currency,  thus, 

£         s.         d.      far. 
4       17       10       3, 

the  language  implies,  that  four  uniti  of  the  lowest  denomination 

20.  To  what  does  one  refer?  What  is  it  called?  To  wliat  does  one 
foot  refer?  'What  is  it  called?— 21.  What  is  an  Abstract  Number?— 
22.  What  is  a  Denominate  Number  ?— 23.  What  is  a  Simple  Number  ? 
When  are  two  numbers  of  the  same  denomination  ?  When  of  different 
denominations ?— 24.  Wliat  is  a  Compound  Denominate  Number? 


F^kMATION    OK   NUMKKliS.  25 

make  one  unit  of  vlie  next  liig-lier;    twelve  of  the  second,  oue 
of  the  tliu'd  ;    and  twenty  of  tlie  third,  i    ■  of  the  fourtli. 

When  figures  arc  written  to  expres.;  Avoirdupois  weight, 
thus, 

T.       cwt      qr.  lb.        oz.  dr. 

27       11       2  24       11  10 

be  language  implies,  that  IC  units  of  the  lowest  denomiuatio 
make  one  unit  of  the  next  higher;    16  of  the  second,  one  o 
the  third ;   25  of  the  third,  one  of  the  fourth ;   4  of  the  fourth, 
oue  of  the  fifth;   and  20  of  the  fifth,  oue  of  the  sixth.     All 
the   other  compound  denominate  numbers   are   formed  on  the 
same  principle :   hence, 

We  pass  from  a  lower  to  the  next  higher  denomination 
by  considering  how  many  units  of  the  lower  make  one  unit 
of  the  next  higher, 

26.  A  Scale  is  a  series  of  numbers  expressing  the  law  of 
relation  between  the  different  units  of  any  number.  There  are 
two  kinds  of  scales  —  Uniform  and  Varying. 

A  Uniform  Scale  is  one  in  which  the  law  of  relation  between 
the  units,  at  any  step  of  the  scale,  is  the  same. 

A  Varying  Scale  is  one  in  which  the  law  of  relation  between 
tlie  units  is  different,  at  different  steps  of  the  scale. 

The  Units  of  a  Scale,  at  any  step,  are  denoted  by  the  num- 
ber of  units  of  the  louver  denomination  which  make  one  unit  of 
the  next  higher. 

25.  When  several  figures  are  written  by  the  side  of  each  other,  what 
docs  the  language  imply  ? 

In  the  English  Currency,  how  many  units  of  the  lowest  denomination 
make  one  t»f  the  next  higher  ?  IIow  many  of  tlie  second  make  one  oi 
th(5  thliii?    IIow  many  of  the  third,  one  of  the  fourth? 

In  Avoirdupois  weight,  how  many  units  of  tlie  lowest  denomination 
make  one  of  the  next  higher?  How  many  of  the  second,  one  of  the 
tliird? 

20.  What  is  a  Scale  ?  IIow  many  kinds  of  scales  are  there  ?  Name 
them.     What  is  a  Uniform  Scale?    What  is  a  Varying  Scale? 


26  FORMATION    OF   NUMBERS. 

Uniform  Scale  of  Tens. 
27.    If  wc  write  a  row  of  I's    thus : 


2  S    .  I'S 

-^a.    -ga.    -g^g    -g 

f-itSfl        JnSS        *-if-iS        fH 

r^Rg  '^'^.S  nnt-i^  'C!.^ 

§gg     §§^     §g2     gg-a 

W^M        W^S        WHH        WHt^ 

111,    111,    111,    111, 

the  language  of  figures  expresses  that  the  unit  of  each  place 
increases  from  right  to  left,  according  to  the  scale  of  tens. 
This  is  called  the  decimal  system  of  numbers,  and  the  scale  is 
uniform. 

United  States  Currency. 

28.  United  States  Currency  affords  an  example  of  a  system 
of  denominate  units,  increasing  according  to  the  scale  of  tens : 
thus, 

^     ^     S     -       • 

a     & 
1      1 

in  which  ten  units  of  any  denomination  make  one  unit  of  the 
next  higher. 

The  dollars  are  denoted  by  $,  and  separated  from  the  dimes, 
cents,  and  mills  by  a  period  (.),  called  the  decimal  point. 

Varying  Scales. 

29.  If  we  write  the  well-known  signs  of  the  English  Cur- 
-ency,  and  place  1  under  each  denomination,  we  shall  have 

£        s.       d.     far. 
1111 

27.  If  several  figures  are  written  by  the  side  of  each  other,  what  does 
the  language  express?  Wliat  name  is  given  to  this  system  of  numbers? 
What  is  the  scale  ? — 28.  How  do  the  different  units  compare  with  each 
other  in  United  States  Currency  ? 


1 

1 

1 

1 

1 

FOliMATION    OF   NUMBERS.  27 

The  signs,  £  s.  d.  and  far.,  denote  the  value  of  the  unit  1 
in  each  denomhuitiou ;  and  they  also  determine  the  relations  be- 
tween the  different  units.  For  example,  this  simple  language 
expresses  the  following  ideas  : 

1st.  That  the  unit  of  the  right-hand  place  is  1  farthing  ;  of 
the  place  next  at  the  left,  1  penny  ;  of  the  next  place,  1  shilling  ; 
of  the  next  place,  1  pound  :   and 

2d.  That  4  units  of  the  lowest  denomination  make  one  unit 
of  the  next  higher ;  12  of  the  second,  one  of  the  third ;  and 
20  of  the  third,  one  of  the  fourth.  Hence,  4,  12,  and  20  are 
the  numbers  which  make  up  the  scale. 

30.  If  we  take  the  denominate  numbers  of  Avou-dupois  weight, 

we  have 

T.    cwt.    qr.      lb.     oz.     dr. 
111111 

in  which  the  units  increase  in  the  following  manner  :  viz.,  count- 
ing from  the  right,  10  units  of  the  lowest  denomination  make 
1  unit  of  the  next  higher  ;  16  of  the  second,  1  of  the  third  ; 
25  of  the  third,  1  of  the  fourth  ;  4  of  the  fourth,  1  of  the  fifth  ; 
20  of  the  fifth,  1  of  the  sixth.  The  scale,  therefore,  for  this 
class  of  denominate  numbers,  varies  according  to  the  above  law 
If  we  take  any  other  class  of  denominate  numbers,  as  the  Troy 
weight,  we  shall  have  a  different  scale,  and  the  scale  will  continue 
to  vary  as  we  pass  from  one  class  of  numbers  to  another.  But 
in  all  the  formations,  we  shall  recognize  the  application  of  the 
same  general  principles. 

31.  There  are,  therefore,  two  general  methods  of  forming  the 
different  systems  of  integral  numbers,  from  the  unit  one.  The 
first  consists  in  preserving  a  uniform  law  of  relation  between 
the  different  units.  If  that  law  of  relation  is  expressed  by  1 0, 
we  have  the  system  of  decimal  or  common  numbers. 

29.  Is  tho  scale  uniform  or  varying  in  the  English  Currency  ?  Namo 
the  units  of  the  scale  at  each  change  of  denomination. — 30.  Name  tlio 
units  of  the  scale,  at  each  step,  in  the  Avoirdupois  weight.  Name  then 
also  in  the  Apothecaries  weight? 


28  INTEGRAL    UNITS. 

The  second  method  consists  in  the  application  of  known, 
though  varying  laws  of  change  in  the  units.  These  changes  in 
the  units,  produce  different  systems  of  denominate  numbers,  each 
of  which  has  its  appropriate  scale, 

Integral  Units  of  Arithmetic. 

32.  The  Integral  Units  of  Arithmetic  are  divided  into  eight 
classes  : 

1.  Units  of  Abstract  Numbers  ; 

2.  Units  of  Currency ; 

3.  Units  of  Length,  or  Linear  Units  ; 

4.  Units  of  Surface  ; 

5.  Units  of  Yolume,  or  Cubic  Units  ; 

6.  Units  of  Weight  ; 

7.  Units  of  Time  ; 

8.  Units  of  Angular  Measure. 

First  among  the  units  of  arithmetic  is  the  abstract  unit  1. 
This  is  the  primary  base  of  all  abstract  numbers,  and  becomes 
the  base,  also,  of  any  denominate  number,  by  merely  naming  the 
particular  thing  to  which  it  is  applied. 

Of  the   Signs. 

33.  The  sign  =,  is  called  the  sign  of  equality.  When  placed 
between  two  numbers,  it  denotes  that  they  are  equal ;  that  is, 
that  each  contains  the  same  number  of  units. 

The  sign  +,  is  called  plus,  which  signifies  more.  When  placed 
between  two  numbers,  it  denotes  that  they  are  to  be  added 
together.     Thus,  3  +  2  =  5. 

The  sign  — ,  is  called  minus,  a  term  signifying  less.  When 
placed  between  two  numbers,  it  denotes  that  the  one  on  tlie 
right  is  to  be  taken  from  the  one  on  the  left.     Thus,  6  —  2  =  4. 

31.  How  many  general  methods  are  there  of  forming  numbers  from 
the  unit  one?  What  is  the  first?  What  is  the  second? — 32.  Into  hov 
many  classes  arc  the  Units  of  Arithmetic  divided  ?    Name  them. 


PROPERTIES  OF  THE   9's.  29 

The  sign  X,  is  called  the  sign  of  multiplication.  When 
placed  between  two  numbers,  it  denotes  that  they  are  to  be  mul- 
tiplied together.  Thus,  12  x  3,  denotes  that  12  is  to  be  multi- 
I>Iied  by  3. 

The  parenthesis  is  used  to  indicate  that  the  sum  or  difference 
)f  two  or  more  numbers  is  to  be  regarded  as  a  single  number. 
Thus,  (2  +  3  +  5)  X  G, 

sliows,  that  the  sum  of  2,  3,  and  5,  is  to  be  multiplied  by  G. 
And  (5  —  3)  X  6, 

denotes  that  the  difference  between  5  and  3,  is  to  be  multiplied 
by  6. 

The  sign  -^,  is  called  the  sign  of  division.  When  placed 
between  two  numbers,  it  denotes  that  the  one  on  the  left  is  to 
be  divided  by  the  one  on  the  right.  Thus,  4  -f-  5,  denotes  that 
4  is  to  be  divided  by  5. 

Properties  of  the  9's. 

34.  In  any  number,  written  with  a  single  significant  figure, 
as,  4,  40,  400,  4000,  &c.,  the  excess  over  exact  D's  is  equal  to 
the  number  of  units  in  the  significant  figure.  For,  any  such 
number  may  be  written  thus, 

4  =  4. 

Also, 40  =  (9      +  1)  X  4, 

400  =  (99    +  1)  X  4, 

4000  =  (999  +  1)   X  4, 

&c.,  &c.,  &c. 

Each  of  the  numbers  9,  99,  999,  &c.,  contains  an  exact  num- 
ber of  9's ;  hence,  when  multiplied  by  4,  the  several  products 
w^ill  contain  an  exact  number  of  9's  ;   therefore, 

03.  What  is  the  sign  of  Equality?  What  is  the  sign  of  Addition? 
Wliat  of  Subtraction?  AVhat  of  Multiplication?  For  what  is  the  pa- 
renthesis used  ?     Wliat  is  the  sign  of  Division  ? 

ol.  Wliat  will  be  the  excess  over  exact  9's  in  any  number  expressed 
by  a  single  significant  figure?  How  may  the  excess  over  exact  9's  be 
found  in  any  number  whate.vcr  ? 


30  REDUCTION  OF 

The  excess  over  exact  9's,  in  each  number,  is  4  ;  and  the 
same  may  he  shown  for  each  of  the  other  significant  figures. 

If  we  write  any  other  number,  as 
6253, 
we  may  read  it,  6  thousands,  2  hundreds,  5  tens,  and  3.    Now, 
the  excess  of  9's  in  the  6  thousands,  is  6  ;  in  2  hundreds,  it  i? 

2  ;  in  5  tens,  it  is  5 ;  and  in  3,  it  is  3  :  hence,  in  them  all,  ii 
is  16,  which  is  one  9,  and  1  over  :  therefore,  t  is  the  excess 
over  exact  9's  in  the  number  6253.     In  Hke  manner, 

The  excess  over  exact  9's,  in  any  number  whatever,  is  found 
by  adding  together  the  significant  figures,  and  rejecting  the 
exact  9' s  from  the  sum. 

Note. — It  is  best  to  reject  or  drop  the  9,  as  soon  as  it  occurs :  thus, 
we  say,  3  and  5  are  8  and  2  are  10;  then,  dropping  the  9,  we  say, 
1  to  6  is  7,  which  is  the  excess ;  and  the  same  for  all  similar 
operations. 

1.  What  is  the  excess  of  9's  in  48t01  ?     In  6t498  ? 

2.  What  is  the  excess  of  9's  in  9412021  ?     In  2704962  ? 

3.  What  is  the  excess  of  9's  in  87049612?     In  4987051? 

REDUCTION. 

35.  Reduction  is  the  operation  of  changing  a  number  from 
one  unit  to  another,  without  altering  its  value. 

36.  Reduction  Descending  is  the  operation  of  changing  a 
Qumber  from  a  greater  unit  to  a  less. 

37.  Reduction  Ascending  is  the  operation  of  changing  a 
number  from  a  less  unit  to  a  greater. 

38.  If  we  have  4  yards,  in  which  the  unit  is  1  yard,  and 
wish  to  change  to  feet,  the  units  of  the  scale  will  be  3,  since 

3  feet  make  1  yard ;  therefore,  the  number  of  feet  will  be 

4  X  3  =   12  feet. 

35.  What  is  Reduction? — 30.  What  is  Reduction  Descending? — o7 
What  is  Reduction  Ascending? 


DENOMINATE   NUMBERS.  31 

If  it  were  required  to  reduce  12  feet  to  inches,  the  units  of  the 
scale  would  be  12,  since  12  inches  make  1  foot :   hence, 

4  yards  =  4  x  3  =  12  feet  =  12  x  12  =  144  inches. 
If,  on  the  contrary,  we  wish  to  change  144  inches  to  feet,  and 
then  to  yards,  we  would  first  divide  by  12,   the  units  of  the 
scale  in  passing  from  inches  to  feet ;  and  then  by  3,  the  unit 
of  the  scale  in  passing  from  feet  to  yards.     Ilence, 

1st.    To  reduce  a  number  from  a  higher  unit  to  a  lower 

Multiiiily  the  units  of  the  highest  denomination  by  the  number 
of  units  in  the  scale,  and  then  add  to  the  product  the  units  of 
the  next  lower  denomination.  Proceed  in  the  same  manner 
through  all  the  denominations  till  the  number  is  brought  to 
the  required  denomination. 

2d.    To  reduce  a  number  from  a  lower  unit  to  a  higher: 

Divide  the  given  number  by  the  number  of  units  in  the  scalCf 
and  set  down  the  remainder,  if  there  be  one.  Divide  the  quo- 
tient thics  obtained,  and  each  succeeding  quotient  in  the  same 
manner,  till  the  number  is  reduced  to  the  required  denomina- 
tion: the  last  quotient,  with  the  several  remainders  annexed, 
will  be  the  answer. 

Examples. 

1.  Kednce  £3  14s.  4d.  to  pence.  We  first  multiply  the  £Z 
by  20,  which  gives  60  shillings.  We  then  add  14,  makmg  14 
shillings :  we  next  multiply  by  12,  and  the  product  is  888  pence : 
to  this  we  add  4d.  and  we  have  892  pence,  which  are  of  the 
same  value  as  £>Z  14s.  4d. 

If,  on  the  contrary,  we  wish  to  change  892  pence  to  pounds, 
shiUings,  and  pence,  we  should  first  divide  by  12 :  the  quotient 
is  14  shillings,  and  4d.  over.  We  next  divide  by  20,  and  the 
quotient  is  i£3,  and  14s.  over:  hence,  the  result  is  £Z  14s.  4d., 
which  is  equal  to  892  pence. 

The  reductions,  in  all  the  denominate  numbers,  arc  made  in 
the  «amc  manner. 


32 


REDUCTION   OF 


2.  Id  £5  5s.,  how  many  shil- 
lings, pence,  and  farlliings  ? 

£5     5s. 
20 

105     5  shilh'ngs  added. 
12 


1260 
4 


5040 

Here  the  reduction  is  from  a 
greater  to  a  less  unit. 

4.    In  34  T.   16  cwt.  3  qr. 
19  lb.,  how  many  pounds? 


3.  In  5040  farthings,  how  many 
pence,  shillings,  and  pounds  ? 

4)  5040  farthings. 
12)  1260  ponce. 
2|0)  10|5    shillings. 
i25  5s. 

In  this  example,  the  reduc- 
tiop  is  from  a  less  to  a  greater 
unit. 


5.    In  69694  lb.,  how  many 
tons,  cwt.,  qr.,  and  lb.  ? 

25)69694 

4)2m  qr.  .      19  lb. 
2|0)69|6  cwt.  .     3  qr. 
34  T.  .    .  16  cwt. 

Ans.  34  T.  16  cwt.  3  qr.  191b. 


69694  lb.  I 

6.  In  $426,  ho^7  many  cents  ?     How  many  mills  ? 

7.  In  36  eagles  8  dollars  and  6  dimes,  how  many  cents  ? 

8.  In  8150  mills,  how  many  dollars  and  cents? 

9.  In  43  eagles  3  dollars  and  5  mills,  how  many  mills  ? 

10.  In  £31  9s.  8d.,  how  many  pence  ? 

11.  In  1569   farthings,  how  many  pounds,   shillings,   pence 
and  f  ir things  ? 

12.  In   IT.    14  cv\t.    1  qr.   20  lb.  Avoirdupois,   how   man} 
pounds? 

13.  In  15445  lb.  Avoirdupois,  how  many  tons,  cwts.,  qrs., 
and  lbs.? 


34 

20 

696 
4 

16  cv,-t.  added. 

2187 
25 

3  qr.  added. 

13954 
55U 

19  lb.  added. 

DENOMINATE   NUMBEliS.  33 

14.  IIow  many  grains  of  silver  in  4  lb.  G  oz.  12  dwt.  and 
T-r.? 

15.  TIow  many  pounds,  ounces,  pennyweights,  and  grains  of 
old  in  T04121  grains? 

IG.  In  5lb  1  3  13  13  2  gr.  Apothecaries'  weight,  how 
n  I  an  j^  grains  ? 

JT.  In  114947  grains,  how  many  pounds,  ounces,  drams, 
.-^(.ruples,  and  grains? 

18.  In  G  yards  2  feet  9  inches,  how  many  inches? 

19.  In  5  miles,  how  many  rods,  yards,  feet,  and  inches? 

20.  In  2730  inches,  how  many  yards,  feet,  and  inches? 

21.  In  56  square  feet,  how  many  square  yards? 

22.  In  355  perches,  or  square  rods,  how  many  acres,  roods, 
and  perches  ? 

23.  In  45G  square  chains,  how  many  acres  ? 

24.  In  3  A.  2  R.  8  P.,  how  many  perches  ? 

25.  In  14  tons  of  round  timber,  how  many  cubic  inches? 
2G.    In  31  cords  of  wood,  how  many  cubic  feet? 

27.  In  5G320  cubic  feet,  how  many  cords? 

28.  In   157  yards  of  cloth,  how  many  nails? 

29.  In  192  ells  Flemish,  how  many  yards? 

30.  In  97  yd.  3  qr.,  how  many  ells  English  ? 

31.  In  4  hlid.  wine  measure,  how  many  quarts  ? 

32.  In  75G0  pints,  wine  measure,  how  many  hogsheads? 

33.  In  7  hogsheads  of  ale,  how  many  pints  ? 

34.  In  74304  half-pints  of  ale,  how  many  barrels? 

35.  In  31  bushels,  dry  measure,  how  many  pints? 

36.  In  2110  pints,  dry  measure,  how  many  bushels?     ' 

37.  In  2  solar  years  of  365  d.  5  h.  48  ra.  48  sec,  each,  how 
many  seconds  ? 

38.  How  many  months,  weeks,  and  days  in  254  days,  reckon- 
ing the  month  at  30  days? 

3* 


3i  ADDITION. 

ADDITION. 

39.  Addition  is  the  operation  of  finding  tlie  sum  of  two  or 
more  numbers. 

The  Sum  of  two  or  more  numbers,  is  a  number  containing  as 
many  units  as  all  the  numbers  taken  together. 

Operations  of  Addition. 
The  operations  of  Addition  depend  on  four  principles,  viz.: 

1.  A  single  number  expresses  a  collection  of  like  units. 

2.  Like  units  alone  can  be  added  together;  that  is,  units  must  Ix 
added  to  units,  tens  to  tens,  dollars  to  dollars,  &c. 

3.  Every  number  expressed  by  two  or  more  figures,  is  the  sum  ot 
its  various  units. 

4.  The  smn  of  sev^cral  numbers  is  equal  to  the  sum  of  all  their  parts. 

1.  What  is  the  sum  of  769  and  487  ?  operation. 
Analysis. — Write  the  numbers,  so  that  the  like  units  7  6  9 

may  fall  in  the  same  column,  thus:  4  8  7 

Sum  of  the  units 16 

Sum  of  the  tens 14 

Sum  of  the  hundreds      ....       1  1 

Entire  sum  12  5  6 

The  example  may  be  done  in  another  way,  thus : 
Set  down  the  numbers  as  before:   then  say,  7  and  9 
are  IG :  set  down  6  in  the  units'  place,  and  the  1  ten  operation. 

under  the  8  in  the  column  of  tens.     Then  say,  1  to  8  7  6  9 

are  9,  and  6  are  15.     Set  down  the  5  in  the  column  of  f  i 

tens,   and  the  1  hundred  in   the  column  of  hundreds. ■ 

We  then  add  the  hundreds,  and  find  their  sum  to  be         t  -^  0  o 
12 :   hence,  the  entire  sum  of  1256. 

Note. — 1.  Observe,  that  units  of  the  same  value  are  always  written 
In  the  same  column. 

2.  When  the  siun  in  any  column  equals  or  exceeds  the  units  of  th 
Bcale  10,  it  produces  one  or  more  units  of  a  higher  order,  wliich  belong 
to  the  next  column  at  the  left.  In  that  case,  write  down  the  excess, 
and  add  the  higher  units  to  the  next  column.  This  is  called  carrying 
to  the  next  column.  The  number  to  be  carried,  should  not,  in  practice 
be  written  under  the  column  at  the  left,  but  added  mentally. 


Al>L»iTiuN. 

(2; 

(3) 

(4) 

85468 

672143 

4783614 

9104 

79161 

504126 

379 

8721 

872804 

94951 


760025 


6160544 


oo 


5.  What  is  the  sum  of  35  dollars  4  dimes  6  cents  5  mWh 
i  dollars  7  mills,  and  97  cents  3  mills? 


Analysis. — Write  the  figures  expressing  units  of 
the  same  value  in  the  same  column,  separating  the 
dollars  from  the  cents  and  mills  by  a  period:  then 
add  the  columns  as  in  simple  numbers. 


OPERATION. 

$35,465 

4.007 

.973 

$40,445 


OPBRATION. 

£        B.     d.     far. 

14    7    8    3 
6  18    9    2 

21     6    6    1 


6.  Let  it  bo  required  to  find  the  sum  of  i£14  7s.  8d.  3far.; 
and  £(j  18s.  9d.  2far. 

Analysis. — Write  the  numbers,  as  before,  so  that  units  of  the  same 
order  sball  fall  in  tlio  same  column.  Beginning  with  tho  lowest  de- 
nomination, we  find  the  smn  to  l)e  5  farthings.  But 
since  4  farthings  make  a  penny,  we  set  down  the 
excess,  1  farthing,  and  carry  one  penny  to  tho  column 
of  pence.  Tho  sum  of  the  pence  then  becomes  18, 
which  is  1  shilling,  and  G  pence  over.     Set  down  the 

0  pence,  and  carry  the  1  shilling  to  the  column  of 
ehillings,    the    sum    of   which   becomes    20 ;    that   is, 

1  pound  and  G  shillings.     Setting  down  the  G  shillings, 

and  carrying  1  to  the  column  of  pounds,  we  find  the  entire  sum  to  bo 
£21  Gs.  Gd.  Ifar. 

Rule. 

I.  Write  the  numbers  so  that  units  of  the  same  value  shall 
fall  in  the  same  column: 

II.  Add  the  units  of  the  lowest  denomination,  and  divide 
their  sum  hy  so  many  as  make  one  unit  of  the  denomination 
next  higher:  set  down  the  remainder,  and  carry  the  quotient 
to  the  next  higher  denomination.  Proceed  in  the  same  man- 
ner through  all  the  denominations,  and  set  down  the  entire 
turn  of  the  last  column 


BQ 


ADDITION. 


Proof. 

40.  The  proof  of  an  operation,  in  Addition,  consists  ia  show- 
ing that  the  answer  ontains  as  many  units  as  there  arc  in  all 
the  numbers  added.     There  are  three  methods  of  proof. 

I.  Begin  with  the  units'  column  and  add,  in  succession,  all 
the  columns  in  an  opposite  direction.  If  the  work  is  right, 
the  residts  will  agn    : 

II.  Divide  the  giccn  numbers  into  parts,  and  add  the  p)0.rls 
separately :  then  add  together  the  partial  sums :  if  the  work  is 
right,  the  residts  will  agree: 

III.  Find  the  excess  of  9's  in  each  number,  and  place  it 
at  the  right  (Art.  34).  Add  these  numbers,  and  note  the  excess 
of  9's  in  their  sum.  This  excess  should  be  equal  to  the  excess 
of  9's  in  the  sum  of  the  numbers. 

Note. — The  third  method  of  proof  applies  only  to  simple  numbers.^ 

1.  What  is  the  sum  of  182t96,  143274,  32160,  and  4t04t? 
and  what  the  proof? 


1st  Method. 
182196 
143214 

32160 

41041 


405211 


2d  Method. 
182196] 
143214  ) 

32160] 

41041) 


326010 


19201 


405211 


89.  "What  is  Addition  ?  What  is  tlie  sum  of  two  or  more  numbers  ? 
On  how  many  principles  do  the  operations  of  Addition  depend  ?  What 
is  the  first  principle  ?  What  the  second  ?  What  the  third  ?  What  the 
fourth  ?    What  is  the  Paile  for  Addition  ? 

40.  How  many  methods  of  Proof  are  there  for  Addition  ?  AYhat  is 
the  process  in  the  first  method?  What  ifi  the  second?  What  in  the 
third  ? 

41.  What  is  the  process  of  reading?  IIow  does  it  differ  from 
epelliug? 


ADDl'l 

'ION. 

37 

3d  Method  of  Proof. 

182706 

•        .        . 

6  excess 

of  D's, 

143274 

. 

3       " 

<( 

32160 

. 

3       " 

<< 

47047 

..7 

4       " 

(( 

405277. 

16       . 

7  excess  of  9's. 

Sum 

Reading. 

41.  The  pupil  should  be  early  taught  to  omit  the  iniermediale 
words  in  tlie  addition  of  columns  of  figures.  Thus,  in  the  above 
example,  instead  of  saying,  7  and  0  are  7 ;  7  and  4  are  eleven ; 
11  and  6  are  seventeen  ;  he  should  simply  say,  seven,  eleven, 
seventeen.  Then,  in  the  column  of  tens,  he  should  say,  five, 
eleven,  eighteen,  twenty-seven ;  and  similarly,  for  the  other 
columns  at  the  left.  This  is  called  reading  the  columns.  Let 
the  pupils  be  often  practised  in  the  readings,  both  separately 
and  in  concert  in  the  class. 


Examples. 

(1) 

(2) 

(3) 

(4) 

94201  - 

80032 

98800 

10304 

46390  • 

4291 

10926 

67491 

37467 

2376 

321 

1324 

4572, 

840 

479 

46 

0. 


VvHiat  is  the  sum  of  1376,  38940,  8471,  23G07,  891? 

6.  What    is    the    sum    of   3480902,    3271,    507321,    91243, 
6001,    169? 

7.  Wliat  i.s  tlie  sum  of  42300,  6000,  347001,  525,  47? 


(S) 

(9) 

(10) 

(11) 

(12) 

dayi 

bushels. 

rods. 

minutes. 

gftllons. 

1276 

47917 

9003 

67321 

760324 

3718 

12031 

1881 

4702 

18720 

9024 

5672 

6035 

1067 

5762 

1028 

728 

3176 

377 

1082 

9131 

47 

2004 

99 

47209 

J8 

ADDITION. 

(13) 

(14) 

'   (15) 

(16) 

(17) 

miles. 

furlongs. 

pouuds. 

dollars. 

casks. 

1600 

47468 

76389 

1602 

40506 

2588 

69012 

1036 

9614 

37219 

9101 

23419 

2671 

4732 

50170 

6793 

15760 

5132 

5675 

32614 

8267 

27900 

6784 

8211 

73462 

4572 

12317 

1672 
(20) 

4455 
(21) 

10001 

(18) 

(19) 

(22) 

a75.365 

$30,365 

$180,000 

$300.40 

$4802.279 

278.056 

28.779 

489.007 

167.275 

1642.107 

420.96 

10.101 

76.119 

18.197 

3026.267 

76.125 

9.08 

16.423 

29.94 

125.093 

41.04 

) 

7.14 

9.011 

10.08 

42.75 

(23 

(24) 

(25) 

(26) 

£.     8. 

d. 

far.     lb. 

oz.   dwt 

fij  I      3 

t).   oz.   dr. 

14  11 

3 

1    174 

11  19 

17  11  7 

17  15  12 

17  18  ] 

LO 

2    75 

10  13 

94  10  6 

29  32  10 

29   7 

6 

0    642 

3  10 

60  9  2 

84  10   9 

42  14  ] 

LI 

3    125 

7   5 

42  3  9 

14   3   7 

17  10 

0 

1     62 

0  16 

12  0  6 

40   9   9 

84   0 

1 

0     39 

1   4 

98  7  5 

76   4   7 

16  19 

8 

2    176 

10  15 

127  1  0 

18  11  15 

(27) 

( 

28) 

(29) 

(30) 

cwt.   qr. 

lb. 

yd. 

qr.  na. 

E.  E.  qr.  na. 

L.   rr.I.  fur. 

174  2 

20 

74 

3  3 

14  4  3 

17  2  7 

820  1 

14 

60 

1  2 

75  1  2 

10  1  4 

136  3 

23 

14 

0  1 

84  3  1 

7  0  6 

47  0 

12 

45 

2  3 

17  2  0 

5  2  3 

84  1 

24 

69 

1  0 

10  0  2 

25  1  0 

90  2 

9 

11 

0  0 

19  1  1 

36  2  2 

7  3 

5 

30 

3  1 

29  3  2 

40  1  0 

ADDITION. 

39 

( 

31) 

( 

;32 

) 

(33) 

(34 

) 

fd. 

ft 

in. 

A. 

E. 

P. 

Tun.  hhd.  gal 

gal.   qt. 

pt. 

174 

1 

11 

77 

3 

39 

714  3  56 

14  3 

1 

2G0 

0 

2 

64 

2 

37 

626  1  48 

74  2 

1 

150 

2 

10 

16 

1 

29 

320  0  29 

96  1 

0 

126 

1 

9 

72 

0 

18 

156  2  31 

47  2 

I 

90 

0 

7 

36 

2 

20 

225  1  42 

22  0 

1 

72 

1 

4 

42 

2 

14 

84  0  17 

65  1 

0 

8 

2 

6 

11 

3 

7 

96  1  34 

19  0 

0 

( 

35  ) 

{' 

36) 

(37) 

(38) 

chal. 

bu. 

qt 

yr. 

T\k. 

da. 

da.    hr.   min. 

qr,   lb. 

oz. 

14 

31 

6 

127 

9 

2 

140  12  27 

44  21 

14 

25 

14 

2 

320 

10 

3 

340  16  40 

14  16 

12 

36 

29 

.7 

146 

8 

1 

227  20  56 

22  10 

11 

42 

24 

3 

75 

6 

0 

102  13  25 

36  19 

7 

39 

32 

1 

70 

11 

2 

67  21  37 

51  13 

9 

56 

19 

5 

54 

7 

1 

14   9  10 

30  22 

11 

14 

20 

4 

27 

4 

3 

10  19  46 

16  15 

15 

39.  The  population  of  the  United  States  and  Territories,  in 
1850,  was  as  follows:  White  population,  19553068;  Free  Col- 
ored population,  434495 ;  Slave  population,  3204313 ;  Indians, 
400674  :   what  was  the  entire  population? 

40.  In  the  j^ear  1850,  the  expenditures  of  the  United  States 
amounted  to  43002168  doHars;  in  1851,  "to  48905879  dollars; 
in  1852,  to  46007893  dollars :  what  were  the  expenditures  of 
the  United  States  for  these  three  years? 

41.  A  man  of  fortune  bequeathed  to  each  of  his  three  sons, 
10492  dollars;  to  each  of  his  two  daughters,  5976  dollars;  to 
bis  wife,  the  remainder  of  his  property,  which  exceeded  tlie 
amount  bequeathed  to  his  children  by  twelve  hundred  dolhirs  : 
find  the  amount  of  his  property. 

42.  A  stage  goes  in  one  day  27  miles  3  furlongs  36  rods; 


40  ADDITION. 

the  ucxt,  32  miles  10  rods;  the  next,  3G  miles  2  furlongs;  tlie 
next,  25  miles  G  furlongs  38  feet :  how  far  did  it  go  in  4  days  ? 

43.  Bought  a  barrel  of  flour  for  eight  dollars  and  seventy- 
five  cents ;  a  ton  of  plaster  for  five  dollars  sixty-two  and  a  half 
cents  ;  a  hat  for  three  dollars  twelve  cents  and  five  mills  ;  fifty 
pounds  of  sugar  for  four  dollars  fifty  cents  and  nine  mills  :  what 
was  the  amount  of  my  bill  ? 

44.  A  lady  bought  a  bonnet  for  $5,375  ;  some  silk  for  $12.03  ; 
some  ribbon  for  80.8t5  ;  a  shawl  for  $9.46  :  what  did  the  whole 
amount  to  ? 

45.  A  wine-merchant  taking  an  invoice  of  his  liquors,  finds 
that  he  has  5  hhd.  36  gaL  2  qt.  of  wine  ;  3  hhd.  15  gal.  1  qt.  1  pt. 
of  rum ;  1  hhd.  2  qt.  of  gin  ;  40  gal.  1  pt.  of  whiskey  :  how  much 
liquor  in  all? 

46.  Tea  was  imported  into  the  United  States,  in  the  year 
1851,  to  the  value  of  $4798005;  in  1852,  $7285817;  in  1853, 
$8224853  :  what  was  the  value  of  the  tea  imported  during  these 
three  years  ? 

47.  The  United  States  exported  tobacco,  in  the  year  1851, 
to  the  amount  of  $9219251;  in  1852,  $10031283;  in  1853, 
$11319319:  what  was  the  entire  value  of  tobacco  exported  in 
these  three  years  ? 

48.  A  man  sold  his  house  and  lot  for  $25840,  which  was 
$3186  less  than  he  gave  for  them;  how  much  did  they  cost 
him? 

49.  A  speculator  bought  three  city  lots  :  for  the  first  he  paid 
$2870.43  ;  for  the  second,  $2346.75  ;  for  the  third,  $1563.82. 
He  sold  the  same  at  an  average  profit  upon  each  of  $476.25: 
what  amount  did  he  receive  for  the  lots? 

50.  "What  is  the  fortune  of  a  merchant  who  has  $79650  in 
real  estate,  $25640  in  merchandise,  $9654  in  furniture  and  librarj', 
$16835  in  stocks,  $12642  in  debts  due  him,  and  $5685  in  cash? 

51.  The  churches  of  the  United  States  and  Territories,  in 
l850,  were  :    Baptists,  9375  ;  Congregationahsts,  1706  ;    Presby- 


ADDITION.  41 

tcrians,  4824  ;  Methodists,  13280  ;  Universalists,  529:  what  was 
the  whole  number  of  churches  belonging  to  these  five  denom- 
inations ? 

52.  In  the  same  year,  the  value  of  the  church  property 
owned  by  the  Baptists  in  the  United  States  and  Territories 
was  $11020855  ;  by  the  Congregationalists,  $7970195  ;  by  the 
Presbyterians,  $14543789  ;  by  the  Methodists,  $14822870  ;  ])y 
the  Universalists,  11752316  :  what  was  the  entire  amount? 

53.  During  the  year  1853,  there  was  coined  in  the  United 
States,  $51888882  of  gold;  $7852571  of  silver;  and  867059 
of  copper  ;  what  was  the  amount  of  money  coined  in  the  United 
States  in  1853? 

54.  A  farmer  sends  to  market  the  following  quantities  of 
butter:  18cwt.  2  qr.  161b.;  1  ton  5  cwt.  211b.;  2  qr.  141b.: 
how  much  did  he  send  in  all? 

55.  A  man  having  84  acres  3  roods  26  perches  of  land,  buys 
120  acres  14  perches  more:  how  much  did  he  then  have? 

56.  Suppose  a  father  divides  his  estate  equally  among  his 
three  sons,  giving  each  twenty-five  thousand  dollars  seven  dimes 
six  cents  and  five  mills :  what  was  the  value  of  the  estate  ? 

57.  A  farmer  has  three  fields  of  grain  :  The  first  yields  1375 
bushels  ;  the  second,  1810  bushels ;  the  third,  1265  bushels  ; 
he  values  his  entire  farm  at  $2975  more  than  the  number  of 
bushels  of  grain  raised  from  these  three  fields  :  what  was  tlie 
value  of  his  farm  ? 

58.  Bought  a  silver  teapot  weigiiing  lib.  6  oz.  12dwt.;  a 
cream-cup,  weighing  10  oz.  18  dwt.  20  gr.;  a  poninger,  weighing 
lloz.  16gr.;  a  dozen  large  spoons,  weighing  lib.  14  dwt. 
1 2  gr. :   what  was  the  weight  of  the  whole  ? 

59.  The  whole  number  of  adults  in  the  United  States  and 
Territories,  over  twenty  years  of  age,  who  could  not  read  and 
write,  in  1850,  was  as  follows :  Of  whites,  males,.  389G64 ; 
females,  573234  ;  free  colored,  males,  407^2  ;  females,  49S00  : 
wh:it  was  tlie  whole  number  ? 


42  ADDITION. 

60.  Caesar  was  murdered  b.  c.  43,  and  Washington  died  a.d. 
1799.  How  many  years  elapsed  between  tlie  death  of  these 
great  men? 

61.  A  forwarding  merchant  had  in  his  store-room,  at  one 
time,  Y500  bushels  of  corn;  12865  bushels  of  wheat;  4680 
bushels  of  oats  ;  3296  bushels  of  barley  ;  and  had  room  enough 
left  to  store  4000  bushels  of  oats  :  how  many  bushels  of  grain 
frould  the  storehouse  hold  ? 

62.  A  man  engaging  in  trade,  had  85164.50  in  cash; 
111810.25  in  goods  ;  $3004  in  notes.  His  net  profits  aver- 
iged  12384.16  a  year,  for  3  years  :  what  was  the  total  value 
jf  the  property  at  the  end  of  the  three  years? 

63.  A  person  paid  two  eagles  for  a  coat ;  four  dollars  and 
?ix  dimes  for  a  hat ;  two  dollars  and  sixty-three  cents  for  a 
rest  ;  eight  dimes  seven  cents  and  five  mills  for  a  knife  :  what 
was  the  amount  of  his  bill? 

64.  From  a  piece  of  cloth,  12  yd.  2  qr.  were  cut  at  one 
lime;  16  yd.  1  qr.  3  na.  at  another,  when  there  were  10  yd. 
1  qr.  1  na.  remaining  :   how  much  was  there  in  the  whole  piece  ? 

65.  A  farmer  purchased  a  plough  for  $91 ;  a  wagon,  for 
$451  ;  a  horse,  for  $110J  ;  a  load  of  hay,  for  $12^  ;  a  harrow, 
for  $31 :   what  was  the  cost  of  the  whole  ? 

66.  If  a  certain  warehouse  be  worth  $12540.31^,  and  one- 
fourth  the  contents  is  valued  at  $5632.108  :  what  is  the  value 
of  the  warehouse  and  the  whole  of  its  contents? 

67.  An  English  gentleman  wishing  to  possess  a  certain  horse, 
offers  in  exchange  another  horse,  valued  at  £25  13s.  6d.,  a 
carriage  valued  at  £15  8s.  9d.  2far.,  and  £IS  in  cash.  The 
offer  was  accepted  :   what  did  he  pay  for  the  horse  ? 

68.  In  1850,  the  State  of  New  York  produced  13121498 
bushels  of  wheat ;  Pennsylvania,  15367691  bushels  ;  Yirginia, 
11212616-busheIs;  Ohio,  14487351  bushels  ;  Missouri,  2981652 
bushels  ;  Illinois,  9414575  bushels  :  what  was  the  whole  num- 
ber of  bushels  produced  by  those  States  in  that  year? 


ADDITION.  43 

69.  A  farmer  sold  his  wheat  for  $825.8t  J  ;  his  barley  for 
$67.12J  ;  his  pork  for  880.10  ;  his  apples  for  $46  :  how  much 
did  he  receive  for  the  whole  ? 

70.  Three  persons  enter  into  copartnership  :  The  first  put  iu 
7825  dollars  capital ;  the  second  put  in  1250  dollars  more  than 
the  first ;  and  the  third  put  in  as  much  as  the  other  two  :  wliti"- 
<vas  the  whole  amount  of  capital  invested? 

71.  A  farmer  raised  in  one  field  240  bush.  3  pk.  2  qt.  Oi 
wheat  ;  in  another,  97  bush.  6  qt. ;  in  anotlier,  42  bush.  1  pk.: 
how  much  did  he  raise  in  the  three  fields  ? 

72.  Add  together  three  hundred  dollars,  ten  eagles,  forty 
dimes,  ninety-six  cents,  seven  mills,  nine  dollars,  forty-seven  cents, 
five  mills,  four  eagles,  three  dollars,  and  nine  dimes. 

73.  What  is  the  sum  of  iE17  10s.  6d.;  ^£25  4s.  lO^d.;  18s. 
6d.  3far.;   iEll  9Jd.;   £1  18s.;    21s.  GJd.? 

74.  A  speculator  bought  a  house  and  lot  for  $4750  ;  he  paid 
$695  for  its  thorough  repair,  and  $165  for  the  introduction  of 
gas  ;  he  then  sold  the  house  at  an  advance  of  $625  above  all 
costs  :    wliat  did  he  receive  for  it  ? 

75.  One  town  is  in  latitude  37°  34'  N.,  and  another  town  in 
latitude  29°  16'  S.:    how  far  apart  are  they  in  latitude? 

76.  A  merchant  bought  4  hogsheads  of  sugar  weighing  re- 
spectively, 19  cwt.  3  qr.,  22  cwt.  1  qr.  18  lb.,  16  cwt.  2  qr. 
121b.,  24  cwt.  Iqr.  191b.;  he  paid  $582.68  for  the  sugar,  and 
$83.24  for  freiglit  and  other  charges  ;  he  sold  the  whole,  and 
gained  $166.48  :   at  what  price  per  lb.  did  he  sell  ? 

77.  The  Deluge,  according  to  Chronology,  occurred  1656  years 
after  the  Creation  ;  the  call  of  Abraham,  427  after  the  Deluge  ; 
the  departure  of  the  Israelites,  430  after  the  call  of  Abraham  ; 
tlie  foundation  of  the  Temple,  479  after  the  departure  of  tlie 
Israelites  ;  the  end  of  the  Captivity,  476  after  the  foundation 
of  the  Temple  ;  and  the  birth  of  Christ,  536  years  after  the 
cad  of  the  captivity  :  how  many  years  from  the  Creation  to  the 
present  time,  it  being  the  year  1864  ? 


44  SUBTBACTION. 


SUBTRACTION. 

42.  SuBTKACTioN  is  tliG  Operation  of  finding  the  difi'erence 
between  two  numbers. 

43.  The  DIFFERENCE  between  two  numbers  is  such  a  number 
as,  added  to  the  less,  will  give  the  greater. 

44.  The  Minuend  is  the  greater  of  the  two  numbers. 

45.  The  Subtrahend  is  the  less  of  the  two  numbers. 

46.  The  Remainder,  or  difference  between  two  numbers, 
is  the  result  of  the  operation. 

47.  When  the  two  numbers  are  equal,  either  may  be  the 
minuend,  and  the  remainder  is  0. 

48.     Principles  which  control  the  operations. 

1.  The  difference  of  two  numbers  added  to  the  less  number,  gives 
the  greater; 

2.  Like  units  alone  can  be  taken  from  each  other; 

3.  The  difference  is  the  same,  if  both  numbers  be  equally  increased. 

49.     Operations  and  Rule. 

1.  From  869  take  327  ;  that  is,  from  8  hundreds  6  tens  and 
9  units,  take  3  hundreds  2  tens  and  1  units. 

Analysis. — Place  the  numbers  so  that  units  of  the  opekation. 

same  order  may  fall  in  the  same  column.     Begin-  8  6  9    min. 

ning  with  the  lowest  order,  we  take  units  from  units ;  3  2  7     sub. 

then  tens  from  tens  ;  then,  hundreds  from  hundreds ;  

and  find  the  remainder  to  be  542.  ^  ^  ^     ^^^' 

42.  Wliat  is  the  difference  between  two  numbers? — 43.  What  is 
subtraction? — 44.  What  is  the  minuend? — 45.  What  is  the  subtra- 
hend?—  4G.  AVliat  is  the  remainder,  or  difference? — 47.  W^lien  is  the 
remainder  0? — 48.  What  are  the  three  principles  that  control  the  opera- 
tions of  Subtraction? — 49.  Give  the  rule  for  finding  the  difference  of 
two  numbers. 


oPKnAno.v. 

i   i 

10                =       H 

G24  in  5   12 

a 

a 

4 

393  =  3     9 

3 

231  =  2     3 

1 

SUliTll  ACTION.  45 

2.  From  llie  number  024  take  393. 

Analysis. — Having  -written  down  tlie  numbers,  we  subtract  3  from  4, 
and  find  a  remainder  1.  At  the  next  step  wc  meet  a  difficulty,  for  we 
cannot  subtract  9  tens  from  2  tens. 

Take  1  hundred  =  10  tens,  from  the  6  hun- 
dreds, and  add  it  to  the  2  tens.  Then,  9  tens 
from  13  tens,  leaves  3  tens,  and  3  hundreds  from 
5,  leaves  2  hundreds,  and  the  remainder  is  231. 

The  remainder  can  bo  found  by  adding,  men- 
tally, 10  to  2  tens,  and  then  saying,  9  from  12, 
leaves  3  tens ;  then  adding  1  to  3  hundreds,  and 
Bay,  4  from  G,  leaves  2  hundreds. 

The  process  of  adding  10  to  a  figure  of  the  minuend,  and  returning 
1  to  the  next  figure  of  the  subtrahend,  at  the  left,  is  called  horroidng. 

3.  From  6  T.  14  cwt.  2  qr.  20  lb.  12  oz.,  take  4  T.  It  cwt. 
1  qr.  21  lb.   10  oz. 

Analysis.— Taking  10  oz.  from  12  oz.,  2  oz. 
remain.  At  the  next  step  we  find  a  diffi- 
culty, for  21  lb.  cannot  be  taken  from  20  lb. 
We  then  take  1  qr.  =  25  lb.  from  the  2  qr. 
and  add  it  to  the  20  lb.,  making  45  lb. ;  then 
say,  21  lb.  from  45  lb.  leaves  24  lb. ;  we  then 
add  1  to  the  next  left-hand  figure  of  the 
subtrahend,  and  say,  2  qr.  from  2  qr.  leaves 
0 ;  then  17  cwt.  from  34  cwt.  leaves  17  cwt., 
and  5  from  C  leaves  1  ton.  1      T?      0     94        9 

Rule. 

I.  Set  down  the  less  number  under  the  greater,  so  that  units 
of  the  same  value  shall  fall  in  the  same  column : 

II.  Begin  wUh  the  units  of  the  lowest  denomination,  an 
ubtract  each  number  from  the  one  above  it: 

III.  When  the  number  of  units  in  any  denomination  of  th 
minuend  is  less  than  in  the  same  denomination  of  the  suUra- 
hend,  suppose  so  many  ^lnits  to  be  added  as  make  one  unit  of 
ihe  next  higher  denomination;  after  which,  add  1  to  the  next 
denomination  of  the  subtrahend,  and  subtract  as  b(fore. 


or 

ERATION. 

T. 

cwt 

qr.      lb. 

oz. 

G 

20 

14 

2     20 

12 

4 

n 

1     21 

1 

10 

1 

17 

0     24 

2 

5 

34 

1     45 

12 

4 

n 

1     21 

10 

40  SUBTRACTION. 

Proof. 
50.   There  are  three  methods  of  proving  Subtraction : 

I.  Add  the  remainder  to  the  subtrahend.  If  the  ivorh  is 
right,  the  sum  will  he  equal  to  the  minuend. 

II.  Subtract  the  remainder  from  the  minuend.  If  the  icorh 
is  right,  the  remainder  will  he  equal  to  the  subtrahend. 

III.  Find  the  excess  of  9's  in  the  minuend,  in  the  subtra- 
hend, and  in  the  remainder.  If  the  work  is  right,  the  excess 
of  9's  in  the  two  last  numbers  will  be  equal  to  the  excess  of 
9's  in  the  first. 

Note. — The  third  method  is  only  applicable  to  simple  numbers. 

What  is  the  difference  between  8U136  and  45302? 

1st  Method.  2d  Method. 

874136  828834  874136 

453021  45302  828834 


828834-J 

874136                                      45302 

3d  Method. 

874136 

,      , 

2  excess  of  9's  in  the  first. 

45302 

. 

.      5       "            "               second. 

828834 

. 

.      6       "             " 

5  +  6  =  11: 

hence, 

the  excess  of  9's  in  the  last  two  nnmberB 

is  2. 

Reading. 

51.   What  is  the  difference  between  426  and  295  ? 

By  the  common  method,  which  is  spelling-,  we  say,         operatioa 
5  from  6,  leaves  1 ;  9  from  13,  leaves  3 ;  1  to  carry  to  2,  4  2  6 

are  3;  3  from  4,  leaves  1.  2  9  5 

By  reading  the  words  which  express  the  final  residt, 
we  make  the  operations  mentally,  and  say,  one,  three,  one. 

60.  How  many  methods  of  proof  are  there  ?    What  is  the  first  ?    What 
the  second?    What  the  third? 

51.  What  is  spelling  of  numbers?    What  is  reading? 

52.  IIow  do  you  find  the  difference  between  two  dates? 


1  3  1 


SUBTRACTION. 


47 


OPKRATION. 

yr.      mo.    dik  hr. 

1855     1     4  15 

1801     3     4  12 

54     4     0  3 


Time  between  Dates. 

52.  What  time  elapsed  between  the  inauguration  of  Mr. 
Jefferson,  March  4th,  12  o'clock  m.,  1801,  and  July  4th,  3  p.m., 
1855? 

Analysis. — Place  the  earlier  date  under  the 
later,  writing  the  number  of  the  year,  reckoned 
from  the  beginning  of  the  Christian  Era,  on  the 
left.  Then,  write  in  the  same  line  the  num- 
ber of  the  month,  reckoned  from  the  first  of 
January ,^he  number  of  the  day,  reckoned  from 

the  first  of  the  month,  the  number  of  the  hour,  reckoned  from  12  at 
night,  and  '•vrite  tlie  number  of  minutes  and  seconds,  if  there  are  any, 
still  at  the  right.     Hence,  to  find  the  time  between  two  date% 

Rule. —  Write  the  earlier  date  under  the  later,  and  subtract 
as  in  compound  numbers  (Art.  49). 

Note, — 1.  In  finding  the  difference  between  dates,  as  in  casting 
Interest,  the  month  is  regarded  as  the  twelfth  part  of  the  year,  and 
as  containing  30  days. 

2.  The  civil  day  begins  and  ends  at  12  o'clock  at  night. 

3.  If  the  earlier  date  is  before  the  Christian  Era,  the  sum  ot  the 
numbers  will  express  the  difference  of  time. 


Examples. 


(1) 

(2) 

(3) 

(4) 

From 

4t256t 

103796 

900372 

1760134 

Take 

1092n 

47217 

167301 

48207 

(5) 

(6) 

(^) 

(8) 

roda. 

dollars. 

millB. 

barrels. 

From 

14623457 

8600000 

162347 

8462 

Take 

32700169 

761820 

56321 

4071 

(9) 

(10) 

(11) 

bushels 

InchM. 

minutes. 

From 

100000 

200763194 

3601789412 

Take 

37214 

2142079 

10031761 

48  SUBTKACTION. 

(13)  (13)  (U) 

cords.  gallons.  pounds. 

From      4200000  8888^77  100000000 

Take         325  9999  23 

(15)  (16)  (17) 

From   I8475.G56  $1000.759       $4871036.008 

Take      32.015  194.375  17362.25 


(18)  (19)  (20) 

£    8.   (1.  far.  T.  cwt  qr.  lb.  yd.   qr.  na. 

From*   25  12  6  2  5  17  3  21  137  1  3 

Take    10  14  3  1  2  9  1  14  19  3  2 


(21)  (22)  (23) 

L.   mi.  fur.  rd.  T.  hbd.  gal.  qt.  pt.  A.    E.   P. 

From   75  2  7  37  14  1  26  2  1  100  2  27 

Take   16  1  4  9  5  3  35  3  1  10  3  30 


(24)  ^  (25) 

bush.      pk.  qt  cord.  ft  In. 

From        1000     3  4  225  42     1242 

Take  25     1  6  100  112       720 


(27)  (28)  (29)  (30) 

Kj    5  3  5   3   e  E.E.   qr.  na.  E.  F.   qr.  no. 

144  10  5  27  4  1  174  3  1  171  1  3 

64  11  7  14  7  2  49  4  2  74  3  2 


(31)  (32)  (33)  (34) 

T.   cwt  qr.  cwt  qr.   lb.  qr.    lb.  oz.  lb.    ot.  dr. 

14  12  2  17  1  21  143  22  12  174  11  10 

1  14  3  14  2  24  74  19  14  30  12  13 


SUBTRACTION. 

•il 

(35) 

(30) 

(37) 

(38) 

A.       B.       P. 

A.         K.       P. 

da.         hr.      mln. 

hr.       njin.     sec 

12     1     32 

112     1     31 

167     21     50 

147     50     61 

1     3     14 

U    2     37 

19     23     64 

94     69     67 

39.  From  $10000  take  $1240.37J. 

40.  From  183701289  take  34627. 

41.  From  17  yr.  9  mo.  1  wk.  6  da.  take  10  yr.  11  mo.  2wk. 
6  da. 

42.  From   144a    71    6  3    13   take   56S)    OJ    7  3    13. 

43.  From  two  eagles  seven  dimes,  take  twelve  dollars  and 
fifty  cents. 

44.  From  forty  dollars  twelve  and  a  half  cents,  take  twenty- 
five  cents  and  seven  mills. 

45.  From  one  eagle  five  dollars  six  dimes  and  ten  cents, 
take  five  dollars  seven  cents  and  four  mills. 

40.  What  sum  of  money  added  to  i£ll  14s.  9Jd.  will  make 
iei33  lis.  OJd.? 

47.  An  apprentice,  who  is  14  years  11  months  3  weeks 
14  hours  68  minutes  old,  is  to  serve  his  master  until  he  is  21 
years  of  age.     How  long  has  he  to  serve? 

48.  The  greater  of  two  numbers  is  seven  millions  three 
hundred  and  four  thousand  and  ten  ;  the  less  is  nine  hundred 
and  fifty  thousand  one  hundred  and  forty.  What  is  theu*  differ- 
ence? 

49.  Mont'  Blanc,  the  highest  mountain  in  Europe,  is  15680 
eet  high  ;  Chimborazo,  the  highest  in  America,  is  21427  feet. 
What  is  the  diflference  in  then*  heights? 

50.  A  man  sold  his  farm  for  seven  thousand  five  hundred 
and  thirty  dollars,  which  was  fifteen  hundred  and  ten  dollars 
more  than  he  gave  for  it.     How  much  did  he  give  for  it? 

61.  The  revenue  collected  at  the  port  of  New  York  for  the 
year  ending  30th  June,  1853,  was  $38289341.58  ;   at  Philadel- 

3 


50  SUBTRACTION. 

phia,  $4537046.16;  at  Boston,  $1203048  52;  at  Baltimore, 
$836431.99.  How  much  more  was  collected  at  the  port  of 
New  York  than  at  the  other  three  ? 

52.  A  man  engaging  in  trade,  found,  at  the  end  of  five  years, 
that  he  had  increased  his  capital  ten  thousand  three  hundred 
and  ten  dollars,  and  that  his  whole  capital  amounted  to  fortj- 
six  thousand  five  hundred  dollars.  How  much  did  he  com- 
mence with? 

53.  The  minuend  exceeds  the  remainder  by  683021,  and  the 
remainder  is  902563.     What  is  the  subtrahend? 

54.  The  amount  of  tea  consumed  in  the  United  States  in  the 
year  1846,  was  16891020  pounds  ;  the  amount  of  coffee, 
124336054  pounds.  How  many  more  pounds  of  coffee  than  of 
tea  were  consumed  ? 

55.  What  number  is  that  to  which,  if  you  add  3126,  the  sum 
will  be  ten  thousand? 

56.  From  a  stack  of  hay  containing  9  T.  3  qr.  20  lb.,  I  sold 
4T.  llcwt.  221b.     How  much  was  then  left? 

5*1.  A  owes  B  £25  ;  after  paying  him  £5  9Jd.,  how  much 
will  he  still  owe  him? 

58.  If  the  distance  from  New  York  to  Liverpool  be  3100 
miles,  what  distance  remains  after  a  ship  has  sailed  800  mi.  5  fur. 
36  rd.? 

59.  Mr.  Jones  bought  a  farm  for  three  thousand  five  hun- 
dred dollars  and  fifty  cents  ;  he  sold  the  same  for  three  thou- 
sand three  hundred  dollars  and  eighty-seven  and  a  half  cents  : 
how  much  did  he  lose  by  the  bargain? 

60.  If  a  lot  of  goods  is  bought  for  $750,  and  sold  for 
$925,871   what  will  be  gained? 

61.  If  I  buy  a  bushel  of  wheat  for  $1.87 J  ;  ten  gallons  o* 
molasses  for  $2.50  ;  five  yards  of  cloth  for  $12.37J  :  how  much 
change  must  I.  receive  back,  if  I  give  in  payment  two  ten- 
dollar  bills  ? 


SUBTRACTION'.  51 

62.  The  population  of  tlie  United  States  in  the  year  1850 
was  23191876,  of  which  3204313  were  slaves:  what  was  the 
free  population? 

63.  England  contains  50922  square  miles  ;  Scotland,  31324 
square  miles  ;  Wales,  7398  square  miles  ;  the  United  States 
contain  2988892  square  miles.  How  many  more  square  miles 
does  the  United  States  contain  than  the  whole  of  Great  Britain  ? 

64.  A  gentleman  of  fortune  owning  an  estate  of  two  hun- 
dred thousand  dollars,  bequeathed  thirty  thousand  dollars  to 
objects  of  charity  ;  twenty-five  thousand  two  hundred  and  fifty 
dollars  to  each  of  his  three  sons  ;  twenty  thousand  five  hundred 
and  seventy-five  dollars  to  his  daughter  ;  and  the  remainder  to 
his  widow.      How  much  did  the  widow  receive? 

65.  The  population  of  New  Orleans,  in  1850,  was  116375  ;  in 
1854  it  was  139190  :  what  was  the  increase  in  four  years? 

66.  Having  deposited  $1500  in  a  bank,  I  drew  out  at  one 
time,  $475.12^ ;  at  another  time,  $300  ;  at  another,  $526,25  : 
how  much  remained  ? 

67.  If  the  Declaration  of  Independence  was  made  at  precisely 
12  o'clock,  on  the  4th  day  of  July,  1776:  how  much  time  will 
have  passed  to  the  4th  day  of  March,  1857^  at  30  minutes  past 
3  o'clock,  p.  M.  ? 

68.  If  I  borrow  $1576  of  a  friend,  and  afterwards  pay  him 
$920.8 7 J  :  how  much  will  I  still  owe  him  ? 

69.  The  first  settlement  made  in  the  United  States  was  at 
Jamestown,  in  Virginia,  May  23,  1607  :  how  many  years,  months, 
and  days,  from  that  time  to  the  4th  of  July,  1856. 

70.  The  sum  of  two  numbers  is  36804,  and  the  greater  is 
ighteen  thousand  nine  hundred  and  twenty-seven  :  what  is  the 
ess  number? 

71.  The  revenue  of  the  United  States  in  the  year  1853  was 
$61337574  ;  the  expenditures,  $54026818  :  how  much  did  the 
revenue  exceed  the  expenditures? 


52  SUBTRACTION. 

72.  From  a  box  of  sugar,  containing  19  cwt.  1  qr.  15  lb., 
there  was  14  cwt.  3  qr.  22  lb.  taken:  how  much  was  left? 

73.  A  ship-builder  sold  a  vessel  for  $50376,  which  cost  him 
$42978:  how  much  did  he  gam? 

74.  A  farmer  sold  his  farm  for  six  thousand  three  hundred 
and  seventy-five  dollars  ;  after  paying  his  debts,  he  has  four 
thousand  and  fifteen  dollars  left :  what  was  the  amount  of  his 
debts  ? 

75.  Gunpowder  was  invented  in  the  year  1330  :  how  many 
years  from  that  time  to  the  year  1856? 

76.  What  number  increased  by  five  thousand  eight  hundred 
and  twenty-nine,  will  become  12103? 

77.  A  speculator  bought  a  quantity  of  flour  for  $2084.50  ;  of 
bacon,  for  $760.87|- ;  of  hops,  for  $1836.25.  He  sold  the  flour 
for  $2375.60;  the  bacon,  for  $912,375;  the  hops,  for  $1750: 
what  did  he  gain  or  lose  on  the  whole  ? 

78.  A  farmer  has  two  pastures,  one  containing  9  A.  3  R. 
32  P.;  the  other,  12  A.  29  P.  He  has  also  two  meadows,  one 
containing  10  A.  2  R. ;  the  other,  15  A.  1  R.  20  P. :  how  much 
more  meadow  than  pasture  has  he  ? 

79.  From  a  pile  of  wood  containing  76  cords  and  6  cord  feet, 
was  taken  at  one  time  20  cords  and  48  cubic  feet ;  at  another 
time,  14  cords  1  cord  foot  and  80  cubic  feet:  how  much  re- 
mained m  the  pile  ? 

80.  A  gentleman  purchased  a  house  worth  $9436 ;  a  carriage 
for  $475.50  ;  a  span  of  horses  for  $840.40.  He  paid  at  one 
time,  $5260 ;  at  another,  $1275,37|- ;  at  another,  $936.42 :  how 
much  remained  unpaid? 

81.  If  a  ship  and  cargo  are  valued  at  $47568.487,  and  the 
cargo  alone  at  $3406.50  :  w^hat  is  the  value  of  the  ship  witliout 
the  cargo  ? 

82.  A  note  on  interest,  dated  July  1st,  1853,  was  to  be  paid 
March  20th,  1856:  how  long  was  it  on  interest? 


SUBTRACTION.  53 

83.  A  gentleman  dying  left  an  estate  of  $50000  ;  after  paying 
his  debts,  which  amounted  to  $5647.50,  he  desired  that  each  of 
his  two  sons  sliould  receive  $15000,  and  his  widow  the  remain- 
der :  how  much  did  the  widow  receive  ? 

84.  Bought  a  hogshead  of  wine,  from  which  was  drawn  32  gal. 
1  qt.  1  pt.  :  how  much  remained  in  the  cask  ? 

85.  The  population  of  Chicago,  in  1850,  was  29963  ;  in  1855 
ic  was  80025 :  what  was  the  increase  in  five  years  ? 

80.  A  land  speculator,  owning  twenty-five  thousand  acres  of 
land,  sells  at  one  time  fifteen  hundred  acres ;  at  another,  four 
thousand  seven  hundred ;  at  another,  twenty-five  hundred  acres ; 
at  another,  seven  hundred  and  fifty  acres :  what  number  of  acres 
has  he  left  ? 

87.  The  latitude  of  New  Orleans  is  29°  57' 30" ;  that  of 
Boston,  42°  21'  23":  what  is  the  difi'erence  in  the  latitude  of 
these  two  places? 

88.  A  person  bought  a  span  of  horses  for  three  hundred 
dollars;  a  carriage  for  $410.50;  a  harness  for  $50,675  ;  he  sold 
the  whole  for  six  hundred  dollars :  did  he  gain  or  lose,  and  how 
much  ? 

89.  The  population  of  Great  Britain  and  its  adjacent  islands, 
in  the  year  1841,  was  18664761;  in  1851  it  was  20930468: 
what  was  the  increase  of  population  in  ten  years  ? 

90.  From  a  piece  of  cloth  containing  47  yards,  a  tailor  cut 
14  yd.  3  qr.  2  na. :  how  much  was  left  ? 

91.  A  tradesman  failing  in  business,  was  indebted  to  A  £105 
19s.  lid.;  to  B,  iei27  10s.  9id. ;  to  C,  £34  18s.  lOd.;  to  D 
£500  19s.;  to  E,  £700  14s.  O^d.  When  this  took  place,  h 
had  in  cash  £50 ;  in  goods,  £350  14s.  9d. ;  in  household  furni 
ture,  £24  lis. ;  and  his  book  accounts  amounted  to  £94  14s.  8d 
If  all  these  were  given  up  to  the  creditors,  how  much  would 
they  lose  ? 


54:  MULTIPLICATION. 


MULTIPLICATION. 

53.  Multiplication  is  the  operation  of  taking  one  number  as 
many  times  as  there  are  units  in  another. 

54.  The  Multiplicand  is  the  number  to  be  taken. 

55.  The  Multiplier  is  the  number  denoting  how  many  limes 
the  multiplicand  is  to  be  taken. 

56.  The  Product  is  the  result  of  the  operation. 

57.  A  Composite  Number  is  one  produced  by  multiplying  two 
or  more  numbers  together.  Thus,  60  is  a  composite  number, 
because  3  x  4  x  5  =  60. 

58.  A  Factor  is  any  one  of  the  numbers  which,  multiplied 
together,  produce  a  composite  number.  Thus,  3,  4,  and  5  are 
factors  of  the  composite  number  60. 

NoTK — 1.  The  product,  after  multiplication,  is  a  composite  number, 
and  tlie  multiplicand  and  multiplier  are  factors  or  producers  of  the 
product. 

2.  Multiplication  is  a  sTiort  method  of  addition.  For,  if  the  multipli- 
cand  be  written  as  many  times  as  there  are  units  in  the  multiplier, 
and  the  numbers  added,  the  sum  will  be  equal  to  the  multiplicand 
taken  as  many  times  as  there  are  units  in  the  multiplier. 

59.    Product  of  two  factors. 

Multiply  the  number  6  by  4. 

6 

Analysis. — ^Write,  in  a  horizontal  line, 
as  many  l*s  as  there  are  units  in  the  mul- 
tiplicand, and  write  as  many  such  lines  as 
there  are  units  in  the  multiplier.  It  is  4  - 
then  evident  that  the  product  will  contain 
as  many  units  as  there  are  in  one  line, 
taken  as  many  times  as  there  are  lines. 

Change  now  the  multiplier  into  the  multiplicand:  that  is,  multiply 
4  by  6. 


r 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

i 

MULTIPLICATION. 


55 


I 


Write,  in  a  vertical  line,  as  many  I's  as  there  are  units  in  the  new 
multiplicand  (4),  and  as  many  vertical  lines  as  there  arc  units  i^  the 
new  multiplier  (C),  when  it  is  again  evident  that  all  the  I's  will  repre- 
sent the  nimiber  of  units  in  the  product.    Hence, 

The  product  of  two  factors  is  the  same,  whichever  factor  is 
used  as  the  multiplier. 

Thus,  3xY  =  tx3  =  21:     also,  6x3  =  3xG  =  18 
\  9x5  =  5x9  =  45:     also,  8x6  =  6x8  =  48 


60.     Product  of  Several  Factors. 
Multiply  the  number  7  by  the  composite  number  6 


2X3. 


;i"' 

=  14 
3 

w 

42 

\v 

21 
_2 

42 


Analysis. — Write  6  horizontal  lines  with  7  units  in  each,  and  it  is 
evident  that  the  product  of  7  X  G  =  42,  will  express  the  number  of 
units  in  all  tho  lines. 

Let  us  first  connect  the  lines  in  seta  of  two  each,  as  at  the  right; 
the  number  of  units  in  each  set  will  then  be  expressed  by  7  X  2  =  14. 
But  there  are  three  sets ;  hence,  the  number  of  units  in  all  the  sets  is 
14  X  3  =  42. 

Again,  if  wo  divide  the  lines  into  sets  of  3  each,  as  at  the  left,  the 
number  of  units  in  each  set  will  be  equal  to  7  X  3  =  21 ;  and  since 
there  are  two  sets,  the  whole  number  of  units  will  be  expressed  by 
21X3  =  42. 


53.  What  is  Multiplication? — 54.  What  is  the  nimiber  to  be  taken 
called? — 55.  What  is  tho  multiplier? — 50.  What  is  the  product? — 
57.  What  is  a  composite  number  ? — 58.  What  is  a  factor  ?  Is  the  prod- 
uct, after  multiplication,  a  composite  number?  Wliat  are  its  factors? 
Why  is  multiplication  a  short  method  of  addition  ? 

59.  In  how  many  ways  may  G  and  4  bo  multiplied  together?  How 
do  the  two  products  compare  with  each  other?  What  principle  does 
tliis  prove? — CO.  If  several  factors  be  multiplied  together,  is  the  prod- 
uct  changed  by  changing  their  order? 


66  MULTirLICATION. 

Since  tlie  product  of  eitlier  two  of  tlie  tliree  factors,  7,  3,  and  2, 
will  ^  be  the  same,  wliicliever  be  taken  for  the  multiplier  (Art.  59), 
and  since  the  same  principle  will  apply  to  that  product  and  to  the 
other  factor,  as  well  as  to  any  additional  factor,  if  introduced,  it  follows 
that. 

The  2^roduct  of  ajiy  numher  of  factors  ivill  be  the  same  in 
vhatever  order  they  are  multiplied, 

61.    When  the  multiplier  is  a  composite  number. 
1.   Multiply  215  by  36  =  3x3x2x2. 
Now,  215x36  =  215x3x3x2x2: 

hence,  from  the  last  principle, 

I.  Separate  the  composite  numher  into  its  factors: 

II,  Multiply  the  multiplicand  and  the  partial  products  by 
the  factors,  in  succession,  and  the  last  product  will  be  the  entire 
product  sought. 

Note. — Any  number  whatever,  as  440,  ending  with  0,  is  a  composite 
number  of  Avhich  10  is  a  factor :  for,  440  =  44  X  10.  If  there  are  two 
O's  on  the  right  of  the  significant  figures,  then  100  is  a  factor;  and 
80  on  for  a  greater  number  of  ciphers.  Hence,  when  there  are  ciphersi 
on  the  right  of  significant  figures,  either  in  the  multiplicand  or  multi- 
plier, or  both 

Ilultiply  the  significant  figures  together,  and  then  annex  thn 
ciphers  to  the  product. 

62.     General  Case  and  Rule. 
1.   Multiply  the  number  62Y  by  214. 

Analysis. — The  multiplicand  627  is  to  be  taken  operation. 

214  times;  that  is,  4  units  times,  1  ten  times,  and  6  2  1 

2  hundred  times.     Taking  it  4  units  times,  gives  2  14 

2508 ;  taking  it  1  ten  times,  gives  627,  of  which  the  „  k  n  o 

lowest  unit  is  1  ten;  hence,  7  is  written  in  the  tens  fi  2  7 

place;    taking  it  2  hundred  times,  gives  1254,  the  12  5  4 

lowest  unit  of  which  is  1  hundred.    Adding,  we  have  ; 

134178  for  the  product.  13  4  17  8 

Note. — ^Whcn  the  multiplier  contains  more  than  one  figure,  the 
product  obtained  by  multiplying  the  multiplicand  by  a  single  figure. 


MULTIPLICATION.  57 

is  called  a  partial  product.  In  the  example,  there  are  tliree  partial 
products,  2508,  627,  and  1254.  The  sum  of  the  partial  products  is  equal 
to  the  product  sought. 

Principles  from  the  Analysis. 

1.  If  units  ho  multiplied  by  units,  the  unit  of  the  product  will 
bo  1 

2.  If  tens  bo  multiplied  by  units,  the  unit  of  the  product  will  be 
J   ten. 

3.  If  hundreds  be  multiplied  by  units,  the  unit  of  the  product  will 
be  1  hundred;  and  so  on. 

4.  If  units  erf"  the  first  order  be  multiplied  by  units  of  a  higher 
order,  the  units  of  the  product  will  be  the  same  as  that  of  the  higher 
order. 

5.  If  units  of  any  order  be  multiplied  by  units  of  any  other  order,  the 
unit  i)f  the  product  will  be  of  an  order  one  less  than  the  sum  of  the 
units  denoting  the  two  orders. 

2.    Multiply  the  compound  number  ^23  8s.  6d.  3far.  by  6. 

Analysis. — Multiplying  3  farthings  by  G,  wo 
have  18  farthings,  equal  to  4d.  and  2fiir. ;  set 
down  the  2far. :  then,  G  times  Gd.  are  30d.,  and 
4  pence  to  carry,  are  40d.,  equal  to  3  shillings 
and  4d.;    then,  6  times  8s.  are  48s.,  and  3s.  to  ^..    , ,      T 

carry,  are  51  shillings,  equal  to  £2  and  11  shil- 
lings ;  then,  6  times  £3  are  £18,  and  £2  to  carry,  are  £20,  which  set 
down. 

Note. — The  unit  of  each  product  will  be  the  same  as  the  unit  oi 
the  multiplicand.  Hence,  for  tho  multiplication  of  all  numbers,  we 
have  the  following 

Rule. 

Multiply  each  order  of  units  in  the  multiplicand,  in  succes- 
sion, beginning  with  the  lowest,  by  each  Jig ure  in  th%multiplier, 
and  divide  each  product  by  so  many  units  as  make  one  wiit 
of  the  next  higher  denomination :  write  down  each  remainder 
under  the  units  of  its  own  order,  and  carry  the  cuotient  to 
the  next  product. 


OPERATION 

£ 

8. 

d. 

far. 

3 

8 

G 

3 

6 

68  MULTli'LICATlON. 

Note. — In  multiplying  United  States  money,  care  must  be  taken 
to  point  off  as  many  places  for  cents  and  mills  as  there  are  in  the 
multiplicand. 

63.    Principles  governing  Multiplication. 

The  principles  governing  the  operations  of  multiplication,  ar 
mainly  the  following  : 

1.  There  are  three  parts  in  every  operation  of  Multiplication :  First, 
the  mvlUplicand ;  second,  the  multiplier;  third,  the  product. 

2.  The  multiplier  is  always  an  abstract  number,  and  shows  how 
many  times  the  multiplicand  is  to  be  taken. 

3.  The  unit  of  the  product  is  always  the  same  as  the  unit  of  the 
multiplicand. 

4.  The  product  is  equal  to  the  sum  of  the  partial  products  which 
arise  from  multiplying  the  multiplicand,  in  succession,  by  each  iigure 
of  the  multiplier. 

5.  If  the  multiplier  is  1,  the  product  will  be  equal  to  the  multi- 
plicand. 

6.  If  the  multiplier  is  greater  than  1,  the  product  will  be  as  many 
times  greater  than  the  multiplicand  as  the  multix)lier  is  greater 
than  1. 

7.  If  the  multiplier  is  less  than  1,  the  product  will  be  such  a  part 
of  the  multiplicand  as  the  multiplier  is  of  1. 


61.  How  do  you  multiply  when  the  multiplier  is  a  composite  number  ? 
What  is  one  factor  of  a  number  ending  in  0  ?  What  is  one  factor  when 
the  number  ends  in  two  O's?  In  three  O's?  &c.  How  do  you  multiply 
such  numbers  together? 

62.  Explain  the  operation  of  multiplying  627  by  214.  Wliat  is  a 
partial  product?  Explain  the  five  principles  which  come  from  this 
analysis.     Give  the  general  rule  for  multiplication. 

63.  What  is  the  first  principle  governing  multiplication  ?  Wliat  the 
second  ?  What  the  third  ?  What  the  fourth  ?  What  the  fiftli  ?  What 
the  sixth  ?    What  the  seventh  ? 

64.  How  many  methods  are  th-crc  of  proving  Multiplication  ?  What 
aro  they? 


MULTll'LlCA'llON. 


69 


Proof. 
64.    There  are  three  methods  of  proving  Multiplication: 

I.  Write  the  multiplier  in  the  place  of  the  multiplicand, 
and  find  the  product,  as  before:  if  the  work  is  right,  the  two 
vroducts  will  be  the  same, 

II.  By  casting  out  the  9's. 

III.  Divide  the  product  by  the  multiplier ^  and  the  result 
will  be  the  multiplicand. 


Multiply    80432 
By         506 

First  Method. 

506 
80432 

506 
80432 

482592 
402160 

1012 
1518 
2024 
4048 

4048 
2024 

40698592 

1518 
1012 

40698592 

40698592 

Note. — Although  wc  generally  begin  the  multiplication  by  the  figure 
of  the  lowest  unit,  yet  wo  may  niultii)ly  in  any  order,  if  we  only  jiro 
serve  the  places  of  the  different  oi-ders  of  units.  In  the  exami)le  at  tlie 
right,  we  began  witli  the  order  of  tens  of  thousands,  or  5th  order. 

Second  Method. 

Let  it  be  required  to  multiply  any  two  numbers  together,  as 
641  and  232. 

Analysis. — We  first  find  the  excess  over 
exact  9's  in  both  factors,  and  then  separate 
each  factor  into  two  parts,  one  of  which  shall 
contain  exact  9's,  and  the  other  the  excess, 
and  unite  tlie  two  by  the  sign  plus.  It  is  now 
required  to  take  639  +  2  =  641,  as  many  times 
as  there  are  units  in  225  +  7  =  232. 

Every  partial  product,  in  this  multiplica- 
tion, contains  exact  9's,  except  14,  which  con- 
tains one  9,  and  5  over;  and  as  the  same  may 
bo  shown  for  any  two  numbers    we  6c*»  that, 


OrERATION. 

641 

=  639 

+ 

2 

232 

=  225 

-f 

7 

4413 

+  14 

450 

3195 

1278 

1278 

148098   +  14 


60 


MULTIPLICATION. 


If  we  find  the  excess  of  9's  in  each  of  two  factors,  and  then 
multipty  them  together,  the  excess  of  9^s  in  their  product  will 
he  equal  to  the  excess  of  9's  in  the  vroduct  of  the  factoids. 


Examples. 

(1) 

(2) 

Ex. 

Multiply        81603     .     .     6 

818321     . 

.     2 

By                    9865     .     .     1 

9814     . 

.     1 

Prod.      864203595     .     .     6 

8080160198     . 

.     2 

3.  By  multiplication,  we  have, 

Ex.  4.             Ex.  8.            Ex.  4. 

Ex.  of  product,  2. 

1285    X    143    X    916    = 

:    1016152880. 

Analysis. — The  excess  of  9's  in  the  product  is  found  by  multiplying 
together  the  excess  of  9's  in  the  factors,  and  casting  out  the  9's  from 
the  product.  The  excess  thus  found  is  equal  to  the  excess  of  9's  in 
the  final  product  of  the  numbers. 


4.   We  have,  also. 


Ex.5. 

869 


Ex.  4.        Ex.  0.  Ex.  0. 

X   49    X   36  =  1532916. 


Note. — When  the  excess  of  9's  in  any  factor  is  0,  tbe  excess  of  9's 
in  the  product  is  always  0. 


Examples. — Simple 

Numbers. 

(1) 

(2) 

(3) 

(4) 

841046 
8 

9801602 
1 

(6) 

510409 
6 

216981 
9 

(5) 

(^) 

103612 
42 

8163021 
126 

90031146 

214 

(8) 

(9) 

(10) 

14168 
235 

894126 
4514 

20034645 
6481 

MULTIPLICATION. 


Gl 


When  the  multiplier  is  a  Composite  Number  (Art.  61). 
(11)  (12)  (13)  (14) 

67537  87456  890462  75046 

12  27  81  72 


(15) 

(10) 

(17) 

(18) 

270456 

815900 

390762 

910000 

460 

6300 

8100 

640000 

When  tho  multiplicand  is  United  States  Currency. 
(19)  (20)  (21)  (22) 

18704.04  $69,476  $481,694  $749,972 

12  36  48  96 


(23) 
$67,492 
104 

(24)                     (25) 
$219,864                $67,492 
140                       320 

(26) 

$890.46 
436 

(27) 
$87,041 
3204 

(28) 
$95,004 
3992 

(29) 
$946,274 
9809 

When  the  multiplicand  is  a  Compound 

Number. 

(30) 

£           8.        d 

20     6     8 
4 

(31) 

T.     qr.      lb.       oz. 

3     3     21     14 

8 

(32) 

yd.     ft     In. 
16      2      9 

7 

(33) 

deg.         /         II 

12     42     55 
9 

(34) 

hhd.    gal.      qt,     pt 

4     42     2     1 

12 

(35) 

E.  F.    qr,    na. 

24     2     3 
24 

62  MULTIPLICATION. 

G-eneral  Examples. 

1.  Multiply  18  T.  2  qr.  161b.  9  oz.  by  48. 

2.  Multiply  Syr.  8  mo.  2  wk.  3  da.  42  m.  by  56. 

3.  Multiply  68  by  tlie  factors  9   and   8   of  the   composite 
number  72. 

4.  Multiply  G7046  by  10  :   also  by  100. 

5.  Multiply  51049  by  100  :   also  by  1000. 

6.  Multiply  4980496  by  1000 :   also  by  10000. 
t.   Multiply  90720400  by  100:   also  by  10000. 

8.  Multiply  74040900  by  1 :   also  by  10. 

9.  Multiply  674936  by  100:   also  by  100000. 

10.  Multiply  478400  by  270400. 

11.  Multiply  367000  by  37409000. 

12.  Multiply  7849000    by   84694000. 

13.  Multiply  89999000  by  97770400. 

14.  Multiply  9187416300  by  274987650000. 

15.  Multiply  86543291213156  by  12637482965. 

16.  Multiply  76729835645873  by  217834569. 

17.  If  it  costs  2479  dollars  to  build  one  mile  of  plank-road, 
how  much  will  it  cost  to  build  25  miles? 

18.  How  far  would  a  vessel  sail  in  9  days,  of  24  hours  each, 
at  the  rate  of  15  miles  an  hour? 

19.  A  man  bought  two  farms,  one  of  125  acres,  at  26  dollars 
an  acre ;  another  of  96  acres,  at  32  dollars  an  acre ;  he  paid 
at  one  time  2500  dollars;  at  another  time,  1725  dollars:  what 
remained  to  be  paid? 

20.  In  9  pieces  of  kersey,  each  containing  14  yd.  3  qr.  2  na., 
how  many  yards? 

21.  What  will  15  gallons  of  wine  cost,  at  5s.  SJd.  per  gallon? 

22.  What  will  be  the  value  of  416  sheep,  at  ^2.48  a  head? 


MULTIPLICATION.  63 

»i3.  Bought  40  barrels  of  flour,  at  $8.75  a  barrel,  and  sola 
them  for  $9.12J  a  barrel:  what  was  the  whole  gain? 

24.  What  is  the  weight  of  1 1  hogslieads  of  sugar,  cacli  weigh- 
ing 7  cwt.  2  qr.  1 8  lb.,  and  what  is  its  value,  at  G  cents  a  pound  ? 

25.  A  merchant  bought  36  pieces  of  broadcloth,  each  contain- 
ing 44  yards,  at  4  dollars  a  yard:  what  did  the  whole  cost? 

26.  A  gentleman,  whose  annual  income  is  $3479,  expends,  fof 
pleasure  and  travelling,  $600;  for  books  and  clothing,  $570; 
for  board  and  other  expenses,  $1200 :  how  much  will  he  have 
saved  in  5  years  ? 

27.  The  number  of  milch  cows  in  the  State  of  Kew  York,  in 
1850,  was  931324  :  what  was  their  value,  at  $18  each? 

28.  If  a  man  travel  20  mi.  5  fur.  16rd.  in  one  day,  how  far 
will  he  travel  in  24  days  ? 

29.  If  a  man  spends  six  cents  a  day  for  cigars,  how  much 
will  he  spend  in  thirty  years,  allowing  three  hundred  and  sixty- 
five  days  to  the  year? 

30.  A  farmer  sold  118  bushels  of  barley  for  62^^  cents  a 
bushel,  and  received  5  barrels  of  flour  at  $9.87^  a  barrel,  and 
the  remainder  in  cash :  how  much  cash  did  he  receive  ? 

31.  Two  persons  start  at  the  same  point  and  travel  in  oppo- 
site directions,  one  at  the  rate  of  34  miles  a  day,  the  other  at 
the  rate  of  28  miles  a  day:  how  far  apart  will  they  be  at  the 
end  of  14  days? 

32.  An  apothecary  sold  8  bottles  of  laudanum,  each  contain- 
ing 10  3   6  3  23  14  gr.:  what  was  the  weight  of  the  whole? 

33.  A  farmer  took  7  loads  of  oats  to  market,  each  load 
having  20  bags,  and  each  bag  containing  2  bush.  3  pk.  6  qt. : 
how  many  bushels  of  oats  did  he  take  to  market? 

34.  The  greatest  number  of  whales  ever  captured  in  the 
northern  seas,  in  one  season,  was  2018.  Estimating  the  oil 
produced  from  each  to  have  been  212  barrels,  wliat  was  the 
amount  of  oil  ? 


64  MULTIPLICATION. 

35.  What  is  the  value  of  an  ox  weighing  1  cwt.  2  qr.  161b., 
at  11  cents  a  pound? 

36.  What  is  the  cost  of  245  hogsheads  of  sugar,  each  weigh- 
ing 984  pounds,  at  1  cents  a  pound? 

31.  Bought  6  loads  of  hay,  each  weighmg  IScwt.  3qr.  21  lb.: 
after  letting  a  neighbor  have  2  tons  15  cwt.  1  qr.  5  lb.,  how 
much  was  there  left  ? 

38.  In  an  orchard  there  are  136  apple-trees,  each  tree  yield- 
ing 17  bushels  of  apples:  how  many  bushels  did  the  whole 
orchard  yield,  and  what  would  they  be  worth,  at  42  cents  a 
bushel  ? 

39.  A  flour  merchant  bought  1845  barrels  of  flour,  at  1  dol- 
lars per  barrel.  He  sold  at  one  tune  628  barrels,  at  9  dollars 
a  barrel ;  at  another  time,  856  barrels,  at  8  dollars  a  barrel  : 
how  many  barrels  had  he  left,  and  at  what  price  could  he  sell 
them,  without  gaui  or  loss  on  the  flour? 

40.  What  are  25  hogsheads  of  sugar  worth,  each  weighing 
872  pounds,  at  6^  cents  a  pound? 

41.  It  is  estunated  that  the  whole  amount  of  land  appropri- 
ated by  the  general  government  for  educational  purposes,  to  the 
1st  of  January,  1854, -was  52770231  acres.  What  was  the  value 
of  this  land,  at  the  government  price  of  one  dollar  and  twenty- 
five  cents  an  acre? 

42.  If  30  men  can  do  a  piece  of  work  in  25  days,  how  long 
will  it  take  one  man  to  do  it? 

43.  A  man  desired  that  his  property  should  be  equally  divided 
among  his  5  children,  giving  each  twenty-seven  hundred  dollars : 
what  was  the  amount  of  his  property  ? 

44.  Bought  9  chests  of  tea,  each  containing  72  pounds,  at 
37^  cents  a  pound :  what  was  the  cost  of  the  whole  ? 

45.  A  merchant  bought  a  box  of  goods  containing  37  pieces, 
each  piece  containing  46  yards,  worth  7  dollars  a  yard :  what 
did  the  box  of  goods  cost? 


MULTIPLICATION.  C5 

46.  A  farm,  consisting  of  127  acres,  was  sold  at  auction  for 
$37,565  an  acre :  what  sum  of  money  did  it  bring  ? 

47.  A  drover  bought  127  head  of  beef  cattle,  at  an  average 
of  39  dollars  per  head;  he  sold  86  of  them  for  43  dollai*s  per 
head :  for  how  much  per  head  must  he  sell  the  remainder,  to 
clear  on  the  first  cost  1246  dollars? 

48.  What  will  75  firkins  of  butter  cost,  each  firkin  weighing 
56  pounds,  at  16  cents  a  pound? 

49.  A  bond  was  given  April  20th,  1850,  and  was  paid  Sept. 
4th,  1856:  what  will  be  the  product,  if  the  time  which  elapsed 
from  the  date  of  the  bond  to  the  time  it  was  paid  be  multiplied 
by  45? 

50.  What  distance  will  a  wheel,  16  feet  8  inches  in  circum- 
ference, measure  on  the  ground,  if  rolled  over  84  times  ? 

51.  What  is  the  difference  between  twice  eight  and  fifty,  and 
twice  fifty-eight? 

52.  How  much  wood  in  4  piles,  each  containing  5  cords, 
6  cord  feet,  and  32  cubic  feet? 

53.  A  man  bought  56  acres  of  land  for  $25  an  acre,  and  94 
acres  for  $32  an  acre  ;  if  he  sells  the  whole  at  $30  an  acre, 
will  he  gain  or  lose,  and  how  much? 

54.  If  12  men  can  build  a  wall  in  16  days,  how  many  men 
will  build  a  wall  nine  times  as  long  in  half  the  time  ? 

55.  A  farmer  sold  4  cows  for  $25.50  each  ;  12  sheep  for 
$2.12J^  each ;  and  3  calves  for  $7.25  each :  what  was  the 
amount  of  the  sale  ? 

56.  If  it  requires  116  tons  of  iron  to  construct  one  mile  of 
railroad,  how  much  would  it  require  to  construct  a  railroad 
from  Albany  to  Buffalo,  it  being  326  miles  ? 

57.  A  merchant  bought  9601b.  of  cheese,  at  9cts.  a  ponnd; 
1181b.  of  butter,  at  121  cts.  a  pound.  He  gave  in  payment, 
12  yd.  of  cloth,  at  $4.75  a  yard;  1861b.  of  sugar,  at  7  cts.  a 
pound,  and  the  remainder  in  cash :  how  much  cash  did  he  pay  ? 


QQ 


MULTIPLICATION. 


58.  How  much  brandy  will  supply  an  anny  of  25,000  men 
for  one  month,  if  each  man  requires  1  gal.  2  qt.  1  pt.  2gi.  ? 

59.  It  is  estimated  that  the  French,  during  the  years  1854 
and  1855,  transported  to  the  Crimea  80000  horses,  and  that 
tOOOO  of  them  were  lost  in  the  same  time.  Supposing  the  first 
cost  of  each  horse  to  be  $100,  and  the  cost  of  transportation 
f  95  per  head,  what  was  the  value  of  the  horses  lost  ?  ^ 

60.  A  man  purchased  a  piece  of  woodland  containing  2t  acres, 
at  39  dollars  per  acre  ;  each  acre  produced  on  an  average  ^0 
cords  of  wood,  which,  being  sold,  yielded  a  net  profit  of  45  cents 
a  cord  :  how  much  did  the  profit  on  the  wood  fall  short  of 
paying  for  the  land? 

Bills  of  Parcels. 

Chicago,  Juno  10,  1857. 

61.  Mr,  John  G.  Smith,  Bought  of  David  Toombs, 


14  pounds 

of  tea. 

at  15  cents,    . 

.      .     $ 

9       " 

coffee, 

u     14       u 

• 

42       " 

sugar, 

"  11   " 

.      , 

3       " 

pepper, 

"    12i  " 

5       " 

chocolate, 

"    56     " 

1      •      • 

12       " 

candles, 

"    16     "         .       , 

►      •      • 

$ 

Received 

payment, 

David  Toombs. 

New  York,  March  20,  1857. 
62.   2Ir.  Jacob  Johns,                Bought  of  George  Bliss  &  Go, 
48  pounds  of  sugar,  at  9 J  cents  a  pound,       .       .     $ 
6  hhds.  of  molasses,  each  containing  63  gallons,  at 
27  cents  a  gallon, 

8  casks  of  rice,  285  lb.  each,  at  5  cts.  a  pound, 

9  chests  of  tea,  86  lb.  each,  at  8*1^  cts.  a  pound, 
4  bags  of  coffee,  each  67  lb.,  at  11  cts.  a  pound. 


Received  payment, 


Geo.  Bliss  &  Co. 


MULTIPLICATION. 


67 


Hartford,  November  21,  1850. 
63.    Gideon  Jones,  Bought  of  Jacob  Thrifty, 

YS  chests  of  tea,  at  $55.65  per  chest,  ...    $ 
251  bags  of  coffee,  100  lb.  each,  at  12J  cts.  per  lb. 
317  boxes  of  raisins,  at  $2.75  per  box, 
1049  barrels  of  shad,  at  $7.50  per  bbl.    . 

76  barrels  of  oil,  32  gal.  each,  at  $1.08  per  gal. 

Amount. 


Received  the  above  in  full. 


Jacob  Thrifty. 


Baltimore,  January  1,  1855. 
64.   fir.  Abel  Wirt,  Bought  of  Timothy  Stout. 

10  yards  of  broadcloth,  at  $4.37^      ....    $ 
75      "  sheeting,       "       .09 

42      "  plaid  prints,  "       .45  .       . 

5  bbl.  Genesee  flour,       "     7.87J      . 
7  pairs  of  boots,  at  $1.60  per  pair, 
18  bushels  of  corn,  at  72  cts.  per  bushel. 


Montreal,  October  16,  1855. 
65.   Mr.  Chas.  Snow,  Bought  of  Fo.se,  Duncan  (^  Co. 

45  yards  of  broadcloth,  at     9s.  6d.         ,,.£&.    d. 
66      " 

16      "  vestings, 

24  lb.  colored  thread, 
72  pair  silk  hose, 
108  yards  carpeting, 


us. 

12s. 

uu. 

9id. 

6s. 

8id. 

5s. 

4d. 

7s. 

5fd. 

Us. 

lOd. 

Received  payment, 


YosE,  Duncan  &  Co 


68  DIVISION. 

DIVISION. 

65.  Division  is  the  operation  of  finding  how.many  times  one 
number  is  contained  in  another  ;  or,  of  dividing  a  number  into 
equal  parts. 

66.  The  Dividend  is  the  number  to  be  divided. 

67.  The  Divisor  is  the  number  by  which  we  divide.  It  is 
the  standard  which  measures  the  dividend  ;  or,  it  shows  into 
how  many  equal  parts  the  dividend  is  to  be  divided. 

68.  The  Quotient  is  the  result  of  division.  It  shows  how 
many  times  the  divisor  is  contained  in  the  dividend,  or  the 
value  of  one  of  the  equal  parts  of  the  dividend. 

69.  The  Remainder  is  what  is  left  after  the  operation.  When 
it  is  0,  the  quotient  is  a  whole  number,  and  the  division  is  exact. 

Numbers  in  Division. 

70.  There  are  always  three  numbers  in  every  division,  and 
sometimes  four  :  First,  the  dividend  ;  second,  the  divisor  ;  third, 
the  quotient ;   fourth,  the  remainder. 

There  are  three  methods  of  denoting  division  ;  they  are  the 
following : 

12  -^  3  expresses  that  12  is  to  be  divided  by  3  ; 
-^  expresses  that  12  is  to  be  divided  by  3  ; 

3)  12     expresses  that  12  is  to  be  divided  by  3. 

When  the  last  method  is  used,  if  the  divisor  does  not  exceed 
12,  we  draw  a  line  beneath  the  dividend,  and  set  the  quotient 
under  it.  If  the  divisor  exceeds  12,  we  draw  a  curved  line  on 
the  right  of  the  dividend,  and  set  the  quotient  at  the  right. 

Kinds  of  Division. 

71.  Short  Division  is  the  operation  of  dividing  when  the 
work  is  performed  mentally,  and  the  results  only  written  down. 
It  is  limited  to  the  cases  in  which  the  divisors  do  not  exceed  12. 

72.  LoNCx  Division  is  the  operation  of  dividing  when  all  the 
work  is  expressed.     It  is  used  when  the  divisor  exceeds  12, 


DIVISION.  69 

73.     Operations  and  Rule. 

1.  Divide  450  by  4. 

Analysis. — The  number  450  is  made  up  of  4  hundreds,  5  tens,  and 

6  units,  each  of  which  is  to  be  divided  by  4.  Dividing  4  hundreds 
by  4,  we  have  the  quotient,  1  hundred :    5  tens  divided 

by  4,  gives  1  ten,  and  1  ten  over :   reducing  this  to  units,  oi'kuation. 

and  adding  in  the  6,  we  have  10  units,  which  contains  4,  4)45  () 

4  times:  hence,  the  quotient  is  114:  that  is,  the  dividend  TT~i 
contains  the  divisor  114  times. 

2.  Divide  £11  8s.  Id.  3far.  by  5. 

Analysis. — Dividing  £11  by  5,  the  quotient  is  £2,  and  £1  remain- 
ing.    Reducing  tliis  to  sliillings,  and  adding  in  the  8,  we  have  28s., 
which,  divided  by  5,  gives  5s.,  and  3s.  over. 
This  being  reduced  to  pence,  and  7d.  added,  operation. 

gives  43d.     Dividing  by  5,  we  have  8d.,  and  ^       ».     d.    far. 

3d.  remainder.    Reducing  3d.  to  farthings,  add-  ^  )  11      8      7     3 

ing  3  farthings,  and  again  dividing  by  5,  gives  2      5      8     3 

the  last  quotient  figure,  3far.     Hence,  £2  5s. 
8d.  ufar.,  is  one  of  the  five  eqna.\  parts  of  the  dividend. 

3.  Divide  Iin2  by  327. 

An^vlysis. — Having  set  down  tlie  divisor  on  the  left  of  the  dividend, 
it  is  seen  that  327  is  not  contained  in  the  first  three  figures  on  the  left, 
which  are  117  hundreds.     But  by  observing 
that  3  is  contained  in  11,  3  times,  and  some-  opkration. 

thing    over,  we  conclude  that  the  divisor  is  327)11772(36 

contained  at  least  3  times  in   the  first  four  ""^ 

figures,  1177  tens,  which  is  a  partial  diddend.  1962 

Set  down  the  quotient  figure  3,  and  multiply  1962 

the  divisor  by  it :  we  thus  get  981  tens,  which 

being  less  than  1177,  the  quotient  figure  is  not  too  great:  we  subtract 
the  981  tens  from  the  first  four  figures  of  the  dividend,  and  find  a 
remainder  196  tens,  which  being  less  than  the  divisor,  the  quotient 
figxire  is  not  too  small.  Reduce  this  remainder  to  imits,  and  add  in 
the  2,  and  we  have  1902. 

As  3  is  contained  in  19,  0  times,  we  conclude  that  the  divisor  is 
contained  in  1902  as  many  as  0  times.  Setting  down  6  in  the  quotient, 
and  multiplying  the  divisor  by  it,  we  find  the  product  to  be  1902. 
Hence,  the  entire  quotient  is  30,  or  the  divisor  is  contained  30  times  in 
the  dividend,  and  30  is  also  one  of  the  327  equal  parts  of  the  dividend.  ^ 


70  DIVISION. 


Rule. 


I.  Beginning  with  the  highest  order  of  units,  take  for  a 
partial  dividend  the  fewest  figures  that  will  contain  the 
divisor:  divide  these  figures  by  it,  for  the  first  figure  of  the 
quotient :  the  unit  of  this  figure  will  be  the  same  as  that  Oj 

he  partial  dividend: 

II.  Multiply  the  divisor  by  the  quotient  figure  so  found, 
and  subtract  the  product  from  the  partial  dividend : 

III.  Reduce  the  remainder  to  units  of  the  next  lower  order, 
and  add  in  the  units  of  that  order  found  in  the  dividend :  this 
gives  a  new  partial  dividend.  Proceed  in  a  similar  manner 
until  units  of  every  order  shall  have  been  divided. 

74.    Directions  for  the  Operations. 

1.  There  are  five  steps  in  the  operation  of  Division :  1st,  To  write 
down  the  numbers;  2d,  To  divide,  or  find  how  many  times;  3d,  To 
multiply ;  4th,  To  subtract ;  5th,  To  bring  down,  to  form  the  partial 
dividends. 

2.  The  product  of  a  quotient  figure  by  the  divisor  must  never  be 
larger  than  the  corresponding  partial  dividend:  if  it  is,  the  quotient 

figure  is  too  large,  and  must  be  diminished. 

♦ 

3.  If  any  one  of  the  remainders  is  greater  than  the  divisor,  the 
quotient  figure  is  too  small,  and  must  be  increased. 

4.  The  unit  of  any  quotient  figure  is  the  same  as  that  of  the  partial 
dividend  from  which  it  is  obtained.  The  pupil  should  always  name 
the  unit  of  every  quotient  figure. 


65.  Wliat  is  Division?— 66.  What  is  the  dividend ?— 67.  What  is  the 
divisor?  What  does  it  show ?— 68.  What  is  the  quotient ?  What  does 
i   show? — 69.  What  is  the  remainder? 

70.  How  many  numbers  are  there  in  every  division  ?  What  are  they  ? 
How  many  signs  of  Division  are  there?     Make  and  name  them. 

71.  What  is  Short  Division?  When  is  it  used?— 72.  Wliat  is  Long 
Division?  When  is  it  used? — 73.  Explain  each  of  the  three  examples. 
Give  the  rule  for  the  division  of  numbers. 


DIVISION.  71 

5.  If  tlie  dividend  and  divisor  are  both  compound  numbers,  reduce 
tliem  to  tlie  same  imit  before  commencing  the  division. 

6.  If  any  partial  dividend  is  less  than  the  divisor,  the  corresponding 
qnotient  figure  is  0. 

7.  When  there  is  a  remainder,  after  division,  write  it  at  the  right 
of  the  quotient,  and  place  the  divisor  under  it. 

75.     Principles  resulting  from  Division. 

1.  When  the  divisor  is  equal  to  the  dividend,  the  quotient  will  be  1. 

2.  When  the  divisor  is  1,  the  quotient  will  be  equal  to  the 
dividend. 

3.  Wlien  the  divisor  is  less  than  the  dividend,  the  quotient  will 
be  greater  than  1.  The  quotient  will  be  as  many  times  greater  than  1, 
as  the  dividend  is  times  greater  than  the  divisor. 

4.  When  the  divisor  is  greater  than  the  dividend,  the  quotient  Vill 
be  less  than  1.  The  quotient  will  be  such  a  part  of  1,  as  the  dividend 
is  of  the  divisor. 

74. — 1.  How  many  steps  are  there  in  division?    Name  them. 

2.  If  a  partial  product  is  greater  than  the  partial  dividend,  what  does 
it  indicate?    What  then  do  you  do? 

3.  What  do  you  do  when  any  one  of  the  remainders  is  greater  than 
the  divisor? 

4.  What  is  the  imit  of  any  figure  of  the  quotient  ?  When  the  divisor 
is  contained  in  simple  units,  what  will  be  the  unit  of  the  quotient  figure  ? 
When  it  is  contained  in  tens,  what  will  be  the  unit  of  the  quotient 
figure?    When  it  is  contained  in  hundreds?    In  thousands? 

5.  If  the  dividend  and  divisor  are  both  compound  numbers,  what 
do  you  do? 

G.  If  any  partial  dividend  is  less  than  the  divisor,  what  is  the  cor- 
responding figure  of  the  quotient? 

7.   When  there  is  a  remainder  after  division,  what  do  you  do  with  it? 

75. — 1.  When  the  divisor  is  equal  to  the  dividend,  what  will  the 
uotient  be? 

2.  When  the  divisor  is  1,  what  wiU  the  quotient  bo? 

3.  When  the  divisor  is  less  than  the  dividend,  how  will  the  quotient 
compare  with  1  ?     How  many  times  will  it  be  greater  than  1  ? 

4.  When  the  divisor  is  greater  than  the  dividend,  liow  will  the 
quotient  compare  with  1  ?    What  part  will  the  quotient  be  of  1  ? 


T2 


DIVH^ION. 


Proofs  of  Division. 
76.   There  are  three  methods  of  proving  division  : 

I.  Multiply  the  divisor  by  the  quotient,  and  add  in  the 
remainder,  if  any:  the  result  should  he  the  dividend. 

II.  Divide  the  dividend,  diminished  by  the  remainder,  if 
any,  by  the  quotient:  the  result  should  be  the  divisor. 

III.  Find  the  excess  of  9's  in  the  divisor  and  in  the  quo- 
tient;  multiply  them  together,  and  note  the  excess  of  O'i;  in 
the  product :  this  should  be  equal  to  the  excess  of  9's  in  the 
dividend,  after  being  diminished  by  the  remai?ider,  if  any. 


1st  Method. 
In  the  last  example,  we  had  11*1*12-^327 
Now,  S2t  X  36  3=  im2. 


36. 


2d  Method. 

Divisor, 

32t,  excess  of  9's     .      ,  •    3 

Quotient, 

36        "             "      .      .       0 

Dividend, 

11172        "             "      .       .       0 

Examples. 

(1) 

(2)                      (3) 

19l3t 

4)  147368          6)  1346840 

I  Product,  0. 


6  )  1050930 


(5) 

6)4'7C898';2 


(6) 
9 )  10324683 


1  )  506321494 


(8) 

I  8 

3)47     19     3 


(9) 

A.       R.      P. 

9)37     3     17 


(10) 

yd.      qr.     na, 

5)47     3     1 


(11) 

$  cts.     ni. 

8)634     15     6 


(12) 

$         cts.      m. 

1)1408   09     0 


(13) 

$  Ct& 

12)802346     16 


DIVISION.  T3 


22.  Divide  $29.25  by  26. 

23.  Divide  $10,125  by  2t. 

24.  Divide  $341.49  by  429. 

25.  Divide  $751.50  by  150. 

26.  Divide  $5111.04  by  108 
21.  Divide  $315  by  $35. 

28.  Divide  $50065  by  $521. 

29.  Divide  $432  by  54. 


14.  Divide  134941644  by  48, 

15.  Divide  8536152  by  36. 

16.  Divide  3361598  by  19. 
11.   Divide  49300  by  125. 
18.   Divide  6411150  by  145. 

9.   Divide  110  by  28. 

20.  Divide  $81,256  by  5. 

21.  Divide  $495,104  by  129. 

30.  Divide  334422198  by  438. 

31.  Divide  114394156  by  1154. 

32.  Divide  41159401184  by  3514. 

33.  Divide  5119481194115  by  45105. 

34.  Divide  4115114931149381  by  11493. 

35.  Divide  611493411549315  by  41143. 

36.  Divide  511943001145  by  31149. 
31.   Divide  1114341149341  by  51143. 

38.  Divide  49311541149315  by  314561. 

39.  Divide  111493115941143  by  511001. 

40.  Divide  6154311495611594  by  618951. 

41.  Divide  1149311418  by  121. 

42.  Divide  11900115108  by  51149. 

43.  Divide  14114931148415  by  123456. 

44.  Divide  129  A.  2  R.  1  P.  by  41. 

45.  Divide  365  da.  6  lir.  by  340. 

46.  Divide  1298  mi.  2  fur.  33  rd.  by  31. 
41.   Divide  95  hlid.  6  gal.  by  120. 

48.    Divide  232  bush.  3  pk.  1  qt.  by  105. 

70.  How  many  mctliods  of  proof  are  tliere  for  division  ?  Wliat  are 
they?  What  is  the  proof  by  multiplication?  What  ia  the  proof  by 
the  9*8? 


74  DIVISION. 

49.  Divide  $18306.25  into  eevon  hundred  and  twonty-five  equal 
parts. 

50.  Bought  1  yards  of  cloth  for  16s.  4d.:  what  did  it  cost 
per  yard? 

61.  A  man  travelled  265  mi.  6  fur.  16  rd.  in  12  days  :  how 
far  did  be  travel  in  one  day? 

62.  If  669  A.  2  R.  23  P.  be  equally  divided  among  9  persons, 
how  much  will  6  of  them  have? 

53.  The  annual  income  of  a  gentleman  is  $10000  :  how  much 
is  that  per  day,  counting  365  days  to  the  year? 

54.  What  number  multiplied  by  0999  will  give  the  product 
987551235? 

55.  A  gentleman  owning  an  estate  of  $*I5000,  gave  one-fourth 
of  it  to  his  wife,  and  the  remainder  was  divided  equally  among 
his  five  children :   how  much  did  each  receive  ? 

56.  The  expenditure  of  the  United  States  for  1853  was 
$54026818  :  how  much  would  that  be  per  day,  allowing  365 
days  to  the  year? 

5t.  If  28  yards  of  cloth  cost  one  hundred  and  thirty-three 
dollars,  what  will  one  yard  cost  ? 

68.  If  I  pay  $63T.50  for  51  yards  of  cloth,  what  is  the  price 
per  yard? 

69*  The  city  of  ^ew  York,  in  1850,  had  104  periodical  prtb- 
lications,  with  an  aggregate  circulation  of  18t4'[600  copies  : 
what  was  the  average  cu-culation  of  each? 

60.  Bought  19  bushels  of  wheat  for  $30.8Y5  :  what  was  the 
cost  of  one  bushel  ? 

61.  How  long  will  9125  loaves  of  bread  last  5  families,  ii 
each  family  consumes  6  loaves  a  day? 

62.  The  product  of  two  numbers  is  1201212012,  and  the 
multiplier  9009  :   what  is  the  multiplicand  ? 

63.  How  many  rings,  each  weighing  4  dwt.  12  gr.,  can  be 
made  from  10  oz.  11  dwt,  12  gr.  of  g-old? 


DIVISION.  75 

64.  If  iron  is  worth  2  cents  a  pound,  how  much  can  be  bought 
for  $67.50  ? 

65.  If  14  sticks  of  hewn  timber  measure  12  T.  38  ft.  118  in., 
how  much  does  each  stick  contain? 

66.  In  1850,  Pennsylvania  manufactured  285702  tons  of  pig 
iron,  and  employed  9285  hands  :  what  was  the  average  product 
of  each  hand? 

07.  The  number  of  college  libraries  in  the  United  States  in 
1850,  was  213,  containing  942321  volumes  :  what  would  be  the 
average  number  of  volumes  in  each? 

68.  Supposing  the  sun  to  make  a  complete  revolution  in  365 
days,  what  is  his  daily  velocity  of  rotation? 

69.  How  many  dozen  spoons  can  be  made  from  31b.  11  oz. 
of  silver,  allowing  15pwt.  16 gr.  to  each  spoon? 

70.  A  gentleman  bought  a  piece  of  cloth,  containing  48  yards, 
at  $3.25  per  yard.  At  what  price  per  yard  must  he  sell  the 
cloth  to  gain^^OO? 

71.  If  i275  18s.  9d.  will  pay  5  men  for  their  weekly  labor, 
what  amount  would  pay  18  men  at  the  same  rate? 

72.  How  many  revolutions  does  the  wheel  of  a  rallcar  make 
in  going  a  distance  of  75  miles,  supposing  the  wheel  to  be 
9  ft.  6  in.  in  circumference  ? 

73.  There  is  a  certain  number,  which  being  divided  by  3,  the 
quotient  multiphed  by  4,  and  6  added  to  the  product,  will  give 
]  8  :   what  is  the  number  ? 

74.  If  a  farm,  containing  512  acres,  costs  $28672,  how  much 
loes  it  cost  per  acre  ? 

75.  If  288120  be  divided  into  432  equal  parts,  what  will  bo 
the  value  of  one  of  the  parts? 

76.  Light  moves  with  immense  rapidity.  It  comes  from  the 
sun  to  the  earth  in  8  minutes,  the  distance  being  96  milhons 
of  miles  :  how  far  does  it  move  in  1  second  ? 


76  CONTRACllONS   IN 


CONTRACTIONS  IN  MULTIPLICATION. 

77.   Contractions  in  Multiplication  are  short  methods  of  find- 
ing products  when  the  multipliers  are  particular  numbers. 

78.    To  multiply  by  25. 

1.   Multiply  the  number  356  by  25. 

Analysis. — If  we  annex  two  ciphers  to  the  mul-  operation. 

tiplicand,  we  multiply  it  by  100  (Art.  Ql):  this  prod-  4)35600 

uct  is  four  times  too  great :  for  the  multiplier  is  but  ~"        ^ 

one-fourth  of  100 ;  hence,  to  multiply  by  25, 

Annex  two  ciphers  to  the  multiplicand,  and  divide  the  re- 
sult by  4. 

Examples. 


1.  Multiply  287  by  25. 

2.  Multiply  184  by  25. 


3.  Multiply  6U1  by  25. 

4.  Multiply  3074  by  25. 


79.     When  the  multiplier  contains  a  fractional  unit. 
1.  What  is  the  product  of  15,  multiplied  by  S}? 

Analysis.— The  multiplicand  is  to  be  taken  3  and  operation. 
one-fifth   times:   taking  it   one-fifth  times,  which   is  1^ 

done  by  dividing  by  5,  gives  3,  which  we  write  in  the  ^5 

unit's  place :  then,  taking  it  3  times,  gives  45,  and  3 

the  sum  48  is  the  product;  hence,  45 

Take  such  a  part  of  the  multiplicand  as  the  ^8  Ans. 

fraction  is  of  l;  then  multiply  by  the  iiitegral  number,  and 
the  sum  of  the  products  will  be  the  required  product. 


Examples 


1.  Multiply  327  by  8|. 

2.  Multiply  23474  by  16|. 

3.  Multiply  34700  by  1271 


4.  Multiply  1272  by  121. 

5.  Multiply  9824  by  2721. 

6.  Multiply  3828  by  73i. 


77.  Wliat  are  contractions  in  Multiplication? — 78.  How  do  you  mul- 
tiply by  25 1 — 79.  How  do  you  multiply  when  the  multiplier  contains  a 
fractional  unit? 


MULTIPLICATION.  77 

80.     To  multiply  by  12 J. 

1.   Multiply  the  number  286  by  12^. 

Analysis.— Since  12J  is  one-eiylith  of  100.  oi'kuahu.n 

Annex  two  ciphers  to  the  rrudtiplicand,  and  — 

diuide  the  result  bij  S.  3  5  7  5 


Examples. 


1.  Multiply  384  by  12J. 

2.  Multiply  47G  by  12^. 


3.  Multiply  14800  by  12^. 

4.  Multiply  670418  by  12^. 


81.    To  multiply  by  33^. 
1.    Multiply  the  number  975  by  33 J. 


OPKRATION. 


Analysis.  —  Annexing   two    ciphers   to    the    mul-  o\Qn^af\ 

tiplicand,  multiplies  it  by  100  :  but  the  multiplier  is  I 

me-third  of  100  :  hence,  3  2  5  0  0 

Annex  two  ciphers,  and  divide  the  result  by  3. 


Examples. 


1.  Multiply  1679252  by  33J. 

2.  Multiply  1480724  by  33i. 


3.  Multiply  10675512  by  33^. 

4.  Multiply  4442172  by  333 J. 


82.     To  multiply  by  125. 
1.   Multiply  the  number  11^  by  125. 


OPERATION. 


I 


Analysis. — Annexing  three  ciphers  to  the  mul-  o  \  i  i  o 

tiplicand,  multiplies  it  by  1000 :  but  125  is  but  ^  )  1  1  25000 

one-eighth  of  one  tliousand :  hence,  14  0  6  2  5 

Annex  three  ciphers,  and  divide  the  result  by  8. 
Examples. 


1.  Multiply  59264  by  125. 

2.  Multiply  17593408  by  125. 


3.  Multiply  1940812  by  125. 

4.  Multiply  140588  by  125. 


80.  Ho(w  do  you  multiply  by  12^?— 81    How  do  you  multiply  by 
Sa^?— 82.  How  do  you  multiply  by'l25? 


78  APPLICATIONS  IN 

Applications. 

83.  The  analysis  of  a  practical  question,  in  Multiplication, 
requires  that  the  multiplier  be  an  abstract  number ;  and  then 
tlie  unit  of  the  product  will  be  the  same  as  the  unit  of  the  mul- 
tiplicand. 

84.  To   find  the  cost  of   several  things,  when  we  know  th 
price  of  one,  and  the  number  of  things 

1.  What  will  six  yards  of  cloth  cost,  at  8  dollars  a  yard? 

Analysis. — Six  yards  of  cloth  will  cost  C  times  as  much  as  1  yard. 
Since  1  yard  of  cloth  costs  8  dollars,  6  yards  will  cost  G  times  8  dol- 
lars, which  are  48  dollars ;  therefore,  G  yards  of  cloth,  at  8  dollars  a 
yard,  will  cost  48  dollars :  hence, 

llie  cost  of  any  number  of  things  is  equal  to  the  price  of  a 
single  thing  multiplied  hy  the  number  of  things. 

We  have  seen  that  the  product  of  two  numbers  will  be  the 
same  (that  is,  contain  the  same  number  of  units),  whichever  be 
taken  for  the  multiplicand  (Art.  59).  Hence,  in  practice,  we 
may  multiply  the  two  factors  together,  taking  either  for  the 
multiplier,  and  the?i  assign  the  proper  unit  to  the  product. 
We  generally  take  the  less  number  for  the  multiphcr. 

85.  To  find  the  cost,  when  the  price  is  an  aliquot  part  of  a 
dollar. 

Note. — For  definition  of  Aliquot  part,  see  Art.  98. 

1.  Find  the  cost  of  45  bushels  of  apples,  at  25  cents  a  bushel. 

Analysis. — If  the  price  were  1  dollar  a  bushel,  the         operation. 
cost  would  be  as  many  dollars  as  there  are  bushels.  4^  45.00 

But  the  price  is  25  cents  =  ^  of  a  dollar ;  hence,  the  .  -        _ 

cost  will  be  one-fourth  as  many  dollars  as  there  are 
bushels ;  that  is,  as  many  dollars  as  4  is  contained  times  in  45,  which 
is  11,  and  1  dollar  over.     This  is  reduced  to  cents  by  adding  tw 
ciphers  ;  then  dividing  again  by  4,  we  have  the  entire  cost :  hence, 

83.  What  does  the  analysis  of  a  practical  question  in  Multiplication 
require?  What  then  follows? — 84.  How  do  you  find  the  cost  of  several 
things  when  you  know  the  price  of  a  single  thing  V — 85.  How  do  you 
find  the  cost  when  the  price  is  an  aliquot  part  of  a  dollar  V 


MULTIPLICATION.  79 

Take  such  a  part  of  the  number  which  denotes  the  amoutit 
of  the  commodity  as  the  j^rice  is  of  1  dollar:  the  result  will 
he  the  cost  in  dollars. 

86.  To  find  the  cxDst,  when  the  price  contains  an  aliquot  part 
of  a  dollar 

Find  the  cost  of  45  bushels  of  wheat,  at  ^2.25  a  bushcL 

AjfALTSis.-— Tho  cost,  at  25  cents  per  45 

bushel,  would  be   $11.25;  and  at   $3  per  __^* 

bushel,  would  be  $90:  hence,  at  $3.25,  it  11.25    at  25  cts. 

la  $101.25.  90*        at  $2.00.* 

Multiply  the  quantity  by  the  aliquot  ""^    ^^^  ' 

part  of  the  dollar :  then  multiply  it  by  the  integral  part  of  the 
price :  the  sum  of  the  products  will  be  the  entire  cost. 

Examples. 

1.  What  would  be  the  cost  of  284  bushels  of  potatoes,  at  50 
cents  a  bushel? 

2.  At  33 J  cents  a  gallon,  what  will  51  gallons  of  molasses 
cost? 

8.  What  cost  112  yards  of  calico,  at  12^  cents  a  yard| 

4.  If  a  pound  of  butter  costs  20  cents,  what  will  1^5  pounds 
cost? 

6.  What  will  576  bushels  of  wheat  cost,  at  $1.50  a  bushel? 

8.  What  will  it  cost  to  dig  a  ditch  129  rods  long,  at  $1.83| 
a  rod? 

T.  At  $1.25  a  barrel,  what  will  96  barrels  of  apples  cost? 

8,  What  will  3  pieces  of  cloth  cost,  each  piece  containing  25 
yards,  at  $1.20  a  yard? 


86.  How  do  you  find  the  coat  when  the  price  contains  an  aliquot 
part  of  a  dollar? 


80  APPLICATIONS  IN 

87.    To  find  the  cost  of  articles  sold  by  the  100  or  1000. 

1.  What  will  544  feet  of  lumber  cost,  at  2  dollars  per  100  ? 

Analysis.— At  2  dollars  a  foot,  the  cost  would  be  544  X  2  =  1088  dol- 
lars ;  but  as  2  dollars  is  the  price  of  100  feet,  it  follows  that  1088  dol- 
ars  is  100  times  the  cost  of  the  lumber ;  therefore,  if  we  divide  1088 
dollars  by  100  (which  is  done  by  cutting  off  two  of  the  right  hand 
figures,  Art.  61),  we  obtain  the  cost 

Note. — Had  the  price  been  so  much  per  1000,  we  should  have 
divided  by  1000:  hence. 

Multiply  the  quantity  by  the  number  denoting  the  price:  if 
the  price  be  by  the  100,  cut  off  two  figures  on  the  right  hand 
of  the  product ;  if  by  the  1000,  cut  o^  three,  and  the  remaining 
figures  will  be  the  answer  in  the  same  denomination  as  the 
price,  which,  if  cents  or  mills,  may  be  reduced  to  dollars. 

Examples. 

1.  What  will  be  the  cost  of  3742  feet  of  tunber,  at  $3.25 
per  100? 

2.  At  $12.50  per  1000,  what  will  5400  feet  of  boards  cost? 

3.  At  89.15  per  hundred,  what  will  be  the  cost  of  1568 
oranges  ? 

4.  What  will  be  the  cost  of  19815  lemons,  at  the  rate  of 
$25  per  thousand? 

6.  Richard  Ames,  Bought  of  John  Maple. 

1215  feet  of  boards,  at    $9.00  per  1000, 


3120 

"     15.25 

11 

115 

"     scantling,    "      8.15 

ti 

1200 

"     timber,       "     12.06 

It 

2550 

"     lathing,       "         .15 

100, 

965 

"      plank,         "       1.12i 
Received  payment, 

ti 

John  Maple. 

87.  How  do  you  find  the  cost  of  articles  sold  by  the  100  or  1000? 


23000 
3450 


MULTIPLICATION.  81 

88.     To  find  the  cost  of  articles  sold  by  the  ton. 
What  is  the  cost  of  640  pounds  of  hay,  at  $11.50  per  ton? 

Analysis. — Since  there  are  2000  opeuation. 

pounds  in  a  ton,  the  cost  of  1000  2)  $11.50 

pounds  will  be  half  as  much  as  of  — ^        .^^  ^^  ^^^^  ^^ 

1  ton :  VIZ.,  $5.75.  Multiply  this  by  640 

the  number  of  pounds  (GIO),  and 
cut  off  throe  places  from  the  right 
(Art.  87),  in  addition  to  the  two 
places  cut  off  for  cents ;  hence,  $3.68000 

3IuUiply  one-half  the  price  of  a  ton  by  the  number  of  pounds^ 
and  cut  off  three  figures  from  the  right  hand  of  the  product 
The  remaining  figures  will  be  the  answer  in  the  same  denomi- 
nation as  the  price  of  a  ton. 

Examples. 

1.  What  will  be  the  cost  of  1575  pounds  of  plaster,  at  $3.84 
per  ton? 

2.  If  one  ton  of  coal  costs  $7.37^,  what  will  be  the  cost  oi 
3496  pounds? 

3.  What  will  1200  pounds  of  hay  cost,  at  $9.40  per  ton  ?  at 
$10.25?   at  $14.60? 

4.  What  will  be  the  cost  of  transportation  of  5482  pounds 
of  iron  from  Pittsburgh  to  New  York,  at  $6.65  per  ton  ? 

5.  What  will  be  the  cost  of  removing  785797  pounds  of 
Btone,  at  $1.87|  per  ton? 

6.  What  will  67418  pounds  of  hay  cost,  at  26  dollars  a  ton? 

7.  What  will  497046  pounds  of  plaster  cost,  at  $9.75  a  ton  ? 

8.  What  is  the  cost  of  9047641  pounds  of  railroad  iron,  at 
$75  a  ton? 


J.  How  do  you  find  the  cost  of  articles  sold  by  the  ton 

1* 


82  CONTKACTIONS   IN 


CONTRACTIONS    IN    DIVISION. 

89.  Coutractions  in  Division   are   short   methods   of  finding 
the  quotient  when  the  divisor  is  a  particular  number. 

90.  By  reversing  the  processes  of  Arts.  78,  80,  81  and  82 
wre  have  the  four  following  rules  : 

1.  To  divide  any  number  by  25: 
Multiply  the  number  by  4,  and  divide  the  product  by  100. 

2.  To  divide  any  number  by  12^: 
Multiply  the  number  by  8,  and  divide  the  product  by  100. 

3.   To  divide  any  number  by  33J,  or  333J,  &c.: 

Multiply  the  number  by  3,  and  divide  the  product  by  100, 
or  1000,  &G. 

4.   To  divide  any  number  by  125: 
Multiply  by  8,  and  divide  the  product  by  1000. 

Examples. 

1.  Divide  OSAO  by  25.  10.  Divide  15851400  by  33i 

2.  Divide  656280  by  25.  11.  Divide  8072400  by  SSJ. 

3.  Divide  7278675  by  25.  12.  Divide  16144800  by  331-. 

4.  Divide  5287215  by  25.  j  13.  Divide  31702800  by  33^. 

5.  Divide  12225  by  12^.  :  14.  Divide  281250  by  125. 

6.  Divide  11925  by  12i.  ;  15.  Divide  6015750  by  125. 

7.  Divide  1760600  by  12^.  ^  16.  Divide  2026875  by  125. 

8.  Divide  67500  by  BSJ.  :  17.  Divide  6080625  by  125. 

9.  Divide  1308400  by  33^.  \  18.  Divide  18047250  by  125. 

89.  What  are  Contractions  in  Division? 

90.  What  rules  do  we  get  by  reversing  tlio  four  last  processes? 


DIVISION.  to 

91.     When  the  divisor  is  a  composite  number. 

1.  How  mauy  feet  and  yards  are  tlicro  in  288  inches? 
Analysis. — Since  there  are  13  inches  In  1  foot,  orHRATioar. 

there  will  ho  as  many  feet  in  288  inches  as  13  is  12)  288 

contained  times  in  288;  viz.,  21  feet,  in  toliicJi  Vie  ~ 

unit  w  1  foot.    Since  3  feet  make  1  yard,  there  vrill  ^ 

be  as  many  yards  in  24  feet  as  3  is  contained  timee  8 

In  24 ;  viz.,  8  yards,  in  which  the  unit  is  1  yard.    We 
have  thus  passed,  by  division,  from  the  unit  1  inch  to  the  unit  1  foc;t, 
and  then   to  the  unit  1   yard;  that  is,  in  each  operation,  we  have 
increased  the  unit  as  many  times  as  there  are  units  in  the  divisor. 

Let  us  now  use  the  same  numbers,  in  a  different  question  : 

2.  If  288  dollars  be  equally  divided  among  36  men,  what  will 
be  the  share  of  each? 

Analysis. — Since  288  dollars  is  to  be  equally  di-  operation. 

vidcd  among  30  men,  each  will  have  as  many  dollars  12)  288 

as  36  is  contained  times  in  288.    Dividing  288  into  13  oVoT 

equal  i>arts,  we  find  that  each  part  is  24  dollars.     If  Z_l_ 

each  of  these  parts  he  now  divided  into  3  equal  parts,  8 

tliere  will  then  be  30  parts  in  all,  each  equal  to  8 
dollars :  here,  tlie  unit  of  the  result  is  the  same  as  thxit  of  tJic  dividend. 
Ilcnco,  we  may  regard  division  under  two  ix)ints  of  view : 

1.  As  a  process  of  reduction,  in  ivhich  the  unit  of  each  suc- 
ceeding dividend  is  increased  as  many  times  as  there  are  units 
in  the  divisor: 

2.  As  a  process  of  separating  a  number  into  equal  parts  ;  in 
ichich  case  the  unit  of  a  part  will  be  the  same  as  that  (f  the 
dividend. 

Hence,  when  the  divisor  is  a  composite  number: 

Divide  the  dividend  by  one  of  the  factors  of  the  divisor ; 
then  divide  the  quotient,  thus  arising,  by  a  second  factor,  and 
so  on,  till  every  factor  has  been  used  as  a  divisor:  the  last  quo- 
tient will  be  the  answer. 

9'1.  Hdt^  do  yoii  divicte  when  the  divisor  is  a  composite  number? 


M  CONTRACTIONS  IN 

Examples. 
Divide  the  following  numbers  by  the  factors  of  the  divisors 


1.  2322    by    6  =  2x3. 

2.  31152  by  24  =  4  X  6. 
8.  19152  by  36  =  6  X  6. 
4.  38592  by  48  =  4  X  12. 


5.  1145592  by  12=    8x9. 

6.  185160  by  96  =  8  X  12 
1.  115716  by  64  =  8x8. 
8.  463104  by  144  =  12  x  12. 


Note. — ^When  there  are  remainders,  after  division,  the  operation  is 
to  be  treated  as  one  of  reduction. 

92.  How  to  find  the  true  remainder. 

1.  Divide  the  number  3611  by  30  =  2  x  3  x  5. 

2)3611 

3)  1835  .      .       1  =  1st  rem.       ...       1 

6 )611  .       .       2  =  2d  rem.             2x2  =  4 

12  2  .       .       1  =  3d  rem.      1x3x2  =_6 

Ans.  122  J  J.                         For  remainder,  11 

i^TALYSis.— Dividing  3071  by  2,  we  have  a  quotient  1835,  and  a 
remainder,  1.  After  the  third  division,  the  quotient  is  122,  and  the  re- 
mainder, 1.     Now,  it  is  plain,  from  the  first  analysis,  that, 

1.  The  Tmit  of  the  first  quotient  is  as  many  times  greater  than  the 
tmit  of  the  dividend,  as  the  divisor  is  times  greater  than  1 ;  and  similarly 
for  all  the  following  quotients. 

2.  The  unit  of  the  first  remainder  is  the  same  as  the  unit  of  the 
dividend ;  and  the  unit  of  any  remainder  is  the  same  as  that  of  the 
corresponding  dividend. 

3.  The  unit  of  any  dividend  is  reduced  to  that  of  the  preceding 
dividend,  by  multiplying  it  by  the  preceding  divisor. 

Hence,  to  find  the  remainder  in  units  of  the  given  dividend 
is  simply  a  case  of  reduction  in  which  the  divisors  denote  the 
units  of  the  scale  :   therefore. 

To  the  first  remainder,  add  the  products  lohich  arise  hy 
mxdtiplying  each  of  the  following  remainders  hy  all  the  pre- 
ceding divisors^  except  its  own :  the  snm  will  he  the  true 
remainder: 


DIVISION. 


85 


Examples. 
Divide  the  following  numbers,  and  find  the  remainders  : 


315  =    7x9x5. 

462  =    3  X  2  X  7  X  11. 

4  X  8  X  9  X  12. 

3x5x7. 


416705  by 

804106  by 

756807  by  3456  = 

8741659  by  105  = 

947043  by  385  =    5  X  7  X  11. 

4704967  by  1155  =  11  x  7  X  5  x  3. 

71874607  by  7560  =    8x7x9x5x3. 

93.     When  the  divisor  is  10,  100,  1000,  Ac. 

1.   Divide  3278  by  1000  =  10  x  10  x  10 

Analysis.— We  divide  3278  by  10,  by 
simply  cutting  off  8,  giving  327  tens,  and 
8  units  remainder.  We  again  divide  by 
10,  by  cutting  off  the  7,  giving  32  hun- 
dreds, and  7  tens  remainder.  We  again 
divide  by  10,  by  cutting  off  the  2,  giving 
a  quotient  of  3  thousands,  and  2  hundreds 
remainder.  The  quotient  then  is  3,  and  a 
remainder  of  2  hundreds,  7  tens,  and  8 
units,  or  278:  hence. 

Cut  off  from  the  right  of  the  dividend  as  many  fir/ures 
as  there  are  cijohers  in  the  divisor^  considering  the  figures 
at  the  left^  the  quotient^  and  those  at  the  rights  the  remainder 


OPERATION. 

10)3  2  7|8 
10)3  2|7      . 
10)  3|2    .     . 
3       .     . 

qj2  7_8_ 


8  rem. 
7  rem. 
2  rem. 


Ans, 


94.    When  any  divisor  contains  significant  figures,  with  one  or 


more  ciphers  at  the  right  hand. 

1.   Divide  875896  by  32000. 

An  Aiorsis.— The  divisor  32000  =  32  X 
1000.  Dividing  by  1000,  gives  a  quotient 
875,  and  896  remainder.  Then  dividing 
by  32,  gives  a  quotient  27,  and  11  remain- 
der, which  gives  the 
hence, 


OPERATION. 

32|000)875|896(27 
64 

235 
224 


Ans 


11896  rem. 
27MS8S- 


86  CONTRACTIONS. 

Cut  off,  hij  a  line,  the  ciphers  from  the  right  of  the  divisor, 
and  an  equal  number  of  figures  from  the  right  of  the  cUvi- 
dend:  divide  the  remaining  figures  of  the  dividend  by  the 
remaining  figures  of  the  divisor,  and  to  the  remainder,  if  any, 
annex  the  figures  cut  off  from  the  dividend,  and  the  result 
will  form  the  true  remainder. 

Examples.  j 

Divide  tiic  following  numbers  : 

1.  1972654  by  420000. 

2.  1752000  by  12000. 

3.  73199006  by  801400. 


4.  11428729800  by  72000. 

5.  36981400  by  146000. 

6.  141614398  by  63000. 


95.     When  the  divisor  contains  a  fraction. 
1.   Divide  856  by  4J. 

AiTALTSis. — There    are    5    fifths    in  1 ;  orERAxioN. 

hence,  in  4  there  are  20  fifths ;  therefore,        3  )  4  2  8  0 

4i  =  21  fifths.    In  the  dividend  856,  there        w  \  -,   .  n  r>  « 

r  *•                     txea            •*  i    *v  *       7)1426.2  rem. 
are  5  times  as  many  fifths  as  units  1 ;  that 

is,  4280  fifths ;  therefore,  the  quotient  is  2  0  3.5  rem. 

4280  divided  by  21,  equal  203lf.    Hence,  ^^^^^   203?-^. 

wlien  the  divisor  contains  a  fraction, 

Ilcduce  the  divisor  and  dividend  to   the  fractional  unit 
of  the  divisor^  and  then  divide  as  in  integral  numbers. 


Examples. 

Find  the  quotients  in  the  following 

examples  : 

1.       3245  ^  161. 

5.   87317  -f-    9f. 

2.      47804  -T-  151 

6.    87906  ~  12f 

3.    870631  ~  141 

7.    95675  -^  15|-. 

4.      37214  -^  511 

8.    71096  ~  17f. 

92.  How  do  you  find  the  true  remainder? 

93.  How  do  you  divide  when  the  divisor  is  10,  100,  &c, 

94.  How  do  you  divide  Avlien  the  divisor  contains  significant  figures, 
«\'ith  ciphers  at  the  right? 

95.  How  do  you  divide  when  tli6  divLsof  Contains  a  fraction? 


PKACTICE.  87 

Applications. 

96.     Division  has  three  applications. 

1.  Given  the  number  of  things  and  their  cost,  to  find  the  price  of 
one  thing. 

2.  Given  the  cost  of  a  number  of  things,  and  the  price  of  one  thing, 
to  find  the  number  of  things, 

3.  To  divide  any  number  of  things,  into  a  given  number  of  equal 
parts. 

Rules. 

I.  Divide  the  number  dcnotiiKj  the  cost  hy  the  number  of 
things:   the  quotient  icill  be  the  price  of  one: 

II.  Divide  the  number  denoting  the  cost  by  the  price  of 
one :   the  quotieiit  icill  be  the  number  of  things : 

III.  Divide  the  number  denoting  the  things  by  the  number 
of  parts  into  which  they  are  to  be  divided:  the  quotient  icill 
be  the  number  i?i  each  2)(trt. 

PRACTICE. 

97.  Pkactice  is  an  easy  and  short  method  of  ajjplying  the 
rules  of  arithmetic  to  questions  which  occur  in  trade  and 
business. 

98.  An  Aliquot  Part  of  a  number  is  any  exact  divisor 
of  it,  whether  integral  or  fractional.  Thus,  3  months  is  an 
aliquot  part  of  a  year,  being  one-fourth  of  it,  and  12^  cents  is 
an  aliquot  part  of  1  dollar,  being  one-eighth  of  it. 

Aliquot  Farts  of  a  Dollar. 

$1  =100  cents, 

i  of  a  dollar  =    50  cents. 

A  of  a  dollar  =  33J-  cents. 

J  of  a  dollar  =    25  cents. 

^  of  a  dollar  =    20  cents. 

96.  How  many  applications  has  division?  Wliat  are  they?  Give 
tlic  rules. 

97.  What  is  Practice ?— 98.  What  is  an  aliquot  part  of  a  number? 


i 

of  a 

dollar 

= 

121 

cents. 

j\ 

of  a 

dollar 

z=z 

10 

cents. 

I'iT 

of  a 

dollar 

— 

6} 

cents. 

A 

of  a 

dollar 

z= 

5 

ceiils. 

-h 

of  a 

cent 

= 

5 

mill>:. 

88 


PllACTICE. 


Aliquot  Parts  of  a  Pound. 

£1  =  20  shillings. 

J  of  a  pound  =  10  shillings, 

i  of  a  pound  =  6s.  8d. 

J  of  a  pound  =  5  shillings. 


4  shillings. 


i  of  a  pound 

•i-  of  a  pound  =  3s.  4d. 

■J  of  a  pound  =  2s.  6d. 

^■2  of  a  pound  =  Is.  8d. 


1  year 

^  of  a  year 

J  of  a  year 


Aliquot  Parts  of  a  Year. 

12  months. 
6  months. 
4  months, 


1  of  a  month  =  15  days, 
i-  of  a  month  =10  days, 
days. 


J  of  a  month 


♦2 


i 

rV 

of  a  year  =  3  months 
of  a  year  =  2  months 
of  a  year   =    1  month. 

Fa 

Month. 

i 
i 

rV 

of  a  month  =  6  days, 
of  a  month  =  5  days, 
of  a  month  =  3  days. 

1.   What  is  the  cost  of  3*16  yards  of  doth,  at  $1  75  a  yard  ? 

Analysis. — At  $1  per  yard, 

$376  =  cost  at  $1  per  yard. 


376  yards  cost  $376.     Separat- 
ing 75  cents  into  50  cents,  an        < 
aliquot  part  of  one  dollar,  and        < 
25  cents,  an  aliquot  part  of  50 
cents,  we  take  one-half  of  $376, 
and  obtain  $188,  the  cost  of 

376  yards  at  50  cents  per  yard.  Since  25  cents  is  one-half  of  50  cents, 
we  take  ^  of  $188,  and  thus  obtain  $94,  the  cost  at  25  cents  per  yard. 
The  sum  $658  gives  the  cost  at  $1.75  per  yard. 


188  =  cost  at  50  cts. 
94  =  cost  at  25    " 
'$658  =  cost  at  $1.15 


2.   What  is  the  cost  of  196  yards  of  cotton,  at  9d.  per  yard? 
A'ALYSis.— 9d.  =  6d.  +  3d. 


The.  cost  of  196  yards  at  ls.= 
196s.  Since  6d.  =  is.,  the  cost 
at  6(1.  =  J  of  196s.  =  98s.  The 
cost  at  3d.  =  ^-  as  much  as  at 
6d. ;  h  of  98s.  =  49s.  Tlie  cost 
at  9d.  =  the  sum  =  147s.  = 
£7  7g. 


196s.  =  cost  at  Is.  per  yard. 


1    1 

6    i 
3    i 

20)  147s.  =  cost  at  9d. 
£1  7s.,  entire  cost. 


98s.  =  cost  at  6d. 
49s.  =  cost  at  3d. 


PRACTICE.  89 

Examples. 

1.  What  is  the  cost  of  425  yards  of  calico,  at  Is.  Gd.  per  yard  ? 

2.  What  is  the  cost  of  415  yd.  of  tape,  at  Id.  Ifar.  per  yard  ? 

3.  What  is  the  cost  of  354  yards  of  cord,  at  IJd.  per  yard  ? 

4.  At  12J  cents  =  $|  a  yard,  what  will  be  the  cost  of  4756 
ards  of  bleached  shirting  ? 

5.  At  2s.  6d.  —  £i  per  pair,  what  will  be  the  cost  of  3154 
pairs  of  gloves  ? 

6.  If  wheat  is  3s.  6d.  a  bushel,  what  will  be  the  cost  of  5320 
bushels  ? 

t.   If  broadcloth  costs  £1  Is.  a  yard,  what  will  be  the  cost 
of  435  yards  ? 

8.  If  linen  is  2s.  6d.  =  2Js.  a  yai:d,  what  will  be  the  cost 
of  660  yards? 

9.  What  will  be  the  cost  of  40  lb.  of  soap,  if  1  pound  costs 
6 J  cents? 

10.  What  will  be  the  cost  of  148  yards  of  cloth,  at  $3.15 
a  yard? 

11.  If  one  bushel  of  apples  cost  62 J  cents,  what  will  be  the 
cost  of  816  bushels? 

12.  What  will  be  the  cost  of  1000  quills,  if  every  5  quills 
cost  1^  cents  ? 

13.  If  1  yard  of  extra-superfine  cloth  costs  $9.50,  what  will 
be  the  cost  of  85  yd.  2qr.? 

14.  What  will  6J  yards  of  cloth  cost,  at  $3.15  a  yard? 

15.  What  will  8J  boxes  of  lemons  cost,  at  $1.25  a  box  ? 

16.  What  will  151  pieces  of  calico  cost,  at  $20.15  a  piece 
11.   If  one  ton  of  iron  costs  $124,  what  will  be  tlic  cost  of 

3T.  15cwt.  2qr.  151b.? 

18.   What  will  be   the   cost   of  350  bushels  of  potatoes,  at 
3s.  6d.  a  bushel,  English  Currency? 


90  LONGlTUDa   AND   TIMS. 


LONGITUDE    AND    TIME. 

99.  The  equator  of  the  earth  is  divided  into  360  equal  parts^ 
?v^hich  are  called  degrees  of  longitude. 

100.  The  sun  apparently  goes  round  the  earth  once  in  24 
iours.     This  time  is  called  a  day. 

Hence,  in  24  hours,  the  sun  apparently  passes  over  360°  oJ 
longitude ;  and  in  1  hour,  over  360°  -^  24  =  15°. 

Since  the  sun,  in  passing  over  15°  of  longitude,  requires  1 
hour,  or  60  minutes  of  time,  in  1  minute  he  will  pass  over 
15  -^  60  =  15'  of  longitude;  and  in  1  second  of  time,  he  will 
pass  over  15'  -f-  60  =  15"  of  longitude  :   Therefore, 

15°  of  longitude  require  1  hour  of  time. 
15'  "  "        1  minute  of  time. 

15"  "  "        1  second  of  time. 

Hence, 

I.  If  the  loJigitude,  expressed  in  degrees^  minutes,  and 
seconds,  be  divided  by  15  =  3x5,  the  quotient  wiU  be  hoursy 
minutes,  and  seconds  of  time. 

II.  If  time,  exj^ressed  in  hours,  minutes,  and  seconds,  be 
multiiMed  ^y  15  =  3  x  5,  the  product  will  be  decrees,  minutes^ 
and  seconds  of  longitude. 

Examples. 

1.   Reduce  45°  31'  45"  of  longitude  to  time. 

5)45°    31'    45" 
Analysis. — We  divide  by  15,  as  in  com-  .  - 

pound  numbera ;  giving  us  3  hr.  2  m.  7  sec 

3  hr.  2  m.  T  sec. 

90.   How  is  the  equator  of  the  earth  divided? 

too.  How  long  is  the  sun  in  apparently  going  round  the  earth? 
What  is  this  time  called?  How  many  degrees  of  longitude  does  the 
sun  pass  over  in  a  day  ?  How  much  in  1  hour  ?  How  much  in  1 
minute?  How  much  in  1  second?  How  is  longitude  reduced  to  time? 
How  is  time  reduced  to  longitude? 


LONGITUDE   AND  TIME.  91 

2.  Reduce  8  hr.  16  m.  40  sec.  of  time,  to  longitude. 

Analysis. — We  multiply  the  seconds,  operation. 

minutes,  and  hours,  each  by  15,  carry-  8  hr.  16  m.  40  sec. 

ing  from  one  to  the  other  as  in  the  l£ 

multiplication  of  compound  numbers.  124°       10'       00" 

3.  If  the  difference  of  time  between  two  places  be  42  m 
16 sec,  what  is  the  difference  of  longitude? 

4.  What  is  the  difference  of  longitude  between  two  places, 
if  the  difference  of  time  is  2  hr.  20  m.  44  sec.  ? 

5.  When  it  is  12  m.  at  New  York,  it  is  llhr.  6  m.  28  sec.  at 
Cincinnati :  what  is  th«ir  difference  of  longitude  ? 

101.    Which  place  has  the  earlier  time. 

When  the  sun  is  on  the  meridian  of  any  place,  it  is  12 
o'clock,  or  noon,  at  that  place.  And  since  the  sun  apparently 
goes  from  east  to  west,  it  will  be  past  noon  for  all  places  at 
the  east,  and  before  noon  for  all  places  at  the  west. 

If,  then,  we  find  the  difference  of  time  between  two  places, 
and  know  the  exact  time  at  one  of  them,  the  corresponding  time 
at  the  other  will  be  found  by  adding  the  difference,  if  that  other 
be  east,  or  by  subtracting  it,  if  west. 

102.  Knowing  the  longitude  of  two  places,  and  the  time  at 
one  place,  to  find  the  corresponding  time  of  the  other. 

1.  The  longitude  of  Albany  is  t3°  42'  west,  and  that  of  Buf- 
falo '18°  55'  west :  what  is  the  time  at  Buffalo  when  it  is  10 
o'clock  A.  M.  at  Albany  ?  , 

Analysis.— The  difference  of  lonffi-  operation. 

tude  is   found  by  subtraction,  and  is  lo      urj» 

5^  13'.    This  difference  is  changed  into  *^     ^^ 

the  time  20  m.  53  sec,  by  dividing  by  15)  5°    13'  diff.  lonjf. 

15.     Since  Buffalo  is  west  of  Albany,  Diff.  time,    20  m.  52  sec. 
this  difference  must  be  subtracted  from 
10  hr..  the   time   at  Albany,  and  the  ^^^^^'     ^  ^'   ^  ^'^^' 

iL-maiudcr  shows  the  time  at  Bufiklo  i >l       ^. 

to  he  9  hr.  39  m.  8  sec.  9  hr  39  ra.  8  sec. 


92  LONGITUDE   AND  TIME. 

Hence  the  following 

Rule. — I.  Reduce  the  difference  of  longitude  to  time: 
II.  Add  the  result  to  the  given  time,  when  the  jylace  at  which 
the  time  is  required  lies  east,  and  subtract  it,  ivhen  west. 

Note. — If  the  longitudes  are  both  east  or  botli  west,  the  difference 
of  longitude  is  found  by  subtraction.  If  one  is  east  and  the  other 
west,  the  difference  of  longitude  is  expressed  by  the  sum. 

Examples. 

1.  The  longitude  of  New  York  is  74°  1'  west,  and  that  of 
Springfield,  Illinois,  89°  33'  west :  what  would  be  the  time  at 
New  York  when  it  is  12  m.  at  Springfield  ? 

2.  The  longitude  of  Philadelphia  is  75°  10'  west,  and  that  of 
New  York  74°  1'  west ;  what  is  the  time  at  Philadelphia,  when 
it  is  3  o'clock  p.  m.  at  New  York  ? 

3.  Washington  is  in  longitude  77°  2' west ;  New  Orleans  in 
89°  2'  west.  When  it  is  9  o'clock  a.  m.  at  Washington,  what  is 
the  time  at  New  Orleans  ? 

4.  The  difference  of  longitude  between  St.  Louis  and  New 
York  is  15°  35'.  In  travelling  from  New  York  to  St.  Louis, 
will  a  watch,  keeping  accurate  time,  be  fast  or  slow  at  St.  Louis, 
and  how  much  ? 

103.  The  time  at  each  of  two  places,  and  the  longitude  of 
one,  being  known,  to  determine  the  longitude  of  the  other. 

1.  New  York  is  in  longitude  74°  1'  west.  In  what  longitude 
is  that  place  whose  tkne  is  10  o'clock  a.  m.,  when  it  is  2  o'clock 
p.  M.  at  New  York  ? , 

101.  What  is  the  time,  at  any  place,  when  the  sun  is  on  the  me- 
ridian? How  will  the  time  then  be  for  any  place  at  the  east?  IIow 
"vfill  it  be  for  any  place  at  the  west?  If  you  have  the  difference  oi 
time  of  two  places,  and  know  the  time  at  one  of  them,  how  do  you 
find  the  time  at  the  other  when  it  is  east?    When  it  is  west? 

102.  Knowing  the  longitude  of  two  places,  and  the  time  at  one,  how 
do  you  find  the  corresponding  time  at  the  other? 

103.  If  the  time  at  each  of  tlie  places  be  known,  and  the  longitude 
of  one,  how  do  you  find  the  longitude  of  the  other? 


14  hr, 

10 

OPEtti 

.  Om. 
0 

^TIOX. 

time 
diff. 
diff. 

>  at  N.  Y. 
at  place. 

4 

0   = 
15 

of  timp. 

60° 

U 

~0'  = 

1 

of  long. 

LONGITUDE   AND  TIME.  93 

AjfALYSiS. — The  difference  of  time 
is  4  hr.,  equal  to  G0°,  T/llich  is  the  dif- 
ference of  longitude.  Since  the  time 
at  New  York  is  later  in  the  day  than 
that  of  the  required  place,  New  York 
must  be  east  of  that  place,  and  the 
longitude  is  found  by  adding  G0°  to 
IP  1  which  gives  134°  1'  west, — the 
required  longitude.  Hence  we  have  134°  1' 
the  following 

Rule. — I.  Reduce  the  difference  of  time  to  difference  of 
longitude : 

II.  Add  the  result  to  the  given  longitude,  when  the  place  at 
which  the  longitude  is  required  has  the  earlier  time,  and  sub- 
tract when  it  has  the  later  time. 

Examples. 

1.  Philadelphia  is  in  longitude  75°  10'  west.  In  what  longi- 
tude is  a  vessel,  whose  chronometer  indicates  11  hr.  30  m.,  a.  m., 
Philadelphia  time,  when  it  is  2  hr.  15  min.,  p.  m.,  on  board  the 
vessel ? 

2.  The  longitude  of  St.  Louis  is  90°  15'  west.  A  person  at 
that  place  observed  an  eclipse  of  the  moon  at  10  hr.  40  m.,  p.  ir. ; 
another  person,  in  a  neighboring  State,  observed  the  same  eclipse 
22  m.  12  sec.  earlier;  what  was  the  longitude  of  the  latter  place, 
and  the  time  of  observation  ? 

3.  Oxford,  in  England,  is  in  longitude  1°  15'  22"  west.  What 
is  the  longitude  of  that  place,  whose  local  tune  is  9  o'clock  p.  m., 
when  the  time  at  Oxford,  as  shown  by  an  accurate  chronome- 
ter, is  101  o'clock,  p.  M.  ? 

4.  In  going  from  London,  whose  longitude  is  0,  to  Orego 
City,  an  accurate  timekeeper  was  found  to  have  gamed  8  hours 
In  what  longitude  is  Oregon  City? 

-  '6.  A  captahi  observed  an  eclipse  of  the  moon  at  11  hr.  18  m. 
15  sec,  p.  M.,  which  was  seen  at  Greenwich,  according  to  the 
Kautical  Almanac,  at  12  hr.  50  m.  19  sec,  p.  m.  In  what  longi- 
tude was  the  vessel? 


94  APPLICATIONS. 


Applications  in  the  Fundpcmental  Rules. 

1.  What  will  it  cost  to  build  a  wall  96  rods  long,  at  $1.33 J 
a  rod? 

2.  A  farmer  wishes  to  put  1066  bush.  2  pk.  of  potatoes  into 
4-74  barrels  :   what  quantity  must  he  put  into  each  barrel  ? 

3.  How  many  barrels  of  apples,  each  containing  2J  bushels, 
can  I  buy  for  $36,  at  45  cents  a  bushel? 

4.  The  quotient  arising  from  a  certain  division  is  1236  ;  the 
divisor  is  375,  and  the  remainder  184  :   what  is  the  dividend  ? 

5.  The  Croton  Water  Works  of  New  York  are  capable  of 
discharging  60000000  gallons  of  water  every  24  hours  :  what 
is  the  average  amount  per  minute? 

6.  The  population  of  the  United  States  in  1850  was  23191876. 
It  is  estimated  that  one  person  in  every  400,  dies  annually  from 
intemperance  :  how  many  deaths  may  be  attributed  annually  to 
this  cause  in  the  United  States? 

7.  If  a  quantity  of  provisions  lasted  25  men  2  mo.  3  wk.  6  da., 
how  long  would  it  have  lasted  10  men? 

8.  If  a  man's  salary  is  $1200  a  year,  and  his  expenses  are 
$640,  how  many  years  will  be  required  to  save  $6720? 

9.  How  long  will  it  take  to  count  20  millions,  at  the  rate  of 
80  per  minute? 

10.  If  3160  barrels  of  pork  cost  $47400,  how  many  barrels 
can  be  bought  for  $11475  ? 

11.  What  will  be  the  cost  of  6  firkins  of  butter,  each  con- 
taining 96  pounds,  at  12 J  cents  a  pound? 

12.  What  will  1000  quills  cost,  at  i  cent  apiece? 

13.  What  will  be  the  cost  of  85i  yards  of  cloth,  at  $9^  a 
yard? 

14.  What  will  be  the  cost  of  1  hhd.  2  gal.  3  qt.  of  brandy, 
at  56-J-  cents  a  quart? 


APIMJCATIOXS.  05 

15.  What  will  b«j  the  cost  of  196  yards  of  cotton  goods,  at 
Is.  6d.  per  yard  ? 

16.  At  23.  8d.  per  bushel,  what  will  1246  bushels  of  oats 
cost? 

17.  If  112  lb.  of  cheese  cost  £2  16s.,  what  is  that  per  pound  ? 

18.  What  will  be  the  cost  of  1426  pounds  of  hay,  at  $9.15 
per  ton? 

19.  How  much  must  I  pay  for  the  transportation  of  3840 
pounds  of  iron,  from  Albany  to  Buffalo,  at  $4.50  per  ton? 

20.  Bought  124  bbl.  of  potatoes,  each  containing  2}  bush., 
at  33 J  cents  a  bushel :   what  was  the  cost  ? 

21.  There  are  three  numbers,  whose  continued  product  is 
16200  ;  one  of  the  numbers  is  25  ;  another,  18  :  what  is  the 
third  number? 

22.  If  1  pwt.  of  gold  is  worth  92  cents,  what  would  be  the 
weight  of  $10059.28  in  gold? 

23.  A  man  sold  his  house  and  lot  for  $4200,  and  took  his 
pay  in  railroad  stock,  at  84  dollars  a  share  :  how  many  shares 
did  he  receive  ? 

24.  A  person  bought  640  acres  of  land,  at  15  dollars  an  acre. 
He  afterwards  sold  160  acres,  at  20  dollars  an  acre  ;  240  acres, 
at  18  dollars  an  acre ;  and  for  the  remainder  he  received  $4560. 
What  was  his  entire  gain,  and  what  did  he  receive  per  acre  on 
the  last  sale? 

25.  A  piece  of  ground,  60  feet  long  and  48  feet  wide,  is  in- 
closed by  a  wall  12  feet  high,  and  2^  feet  thick  :  how  many 
cubic  feet  in  the  wall  ? 

26.  What  will  be  the  cost  of  transportation  from  Montreal 
vo  Boston  of  325640  feet  of  lumber,  at  $2.37J  per  thousand? 

2T.  Bought  684  pounds  of  hay,  at  $12.40  a  ton  :  what  did  it 
cost  me? 

28.   At  $2.12-J  a  hundred,  what  will  186  feet  of  lumber  cost  ? 


96  APPLICATIONS. 

29.  How  many  shingles  will  it  require  to  cover  the  roof  of  a 
building  40  feet  long  and  26  feet  wide,  with  rafters  IG  feet  long, 
allowing  one  shingle  to  cover  24  square  inches  ? 

30.  If  14  lb.  8  oz.  12  pwt.  3  gr.  of  silver  be  made  into  9  tea- 
pots of  equal  weight,  what  will  be  the  weight  of  each  ? 

31.  A  man  bought  320  barrels  of  flour  for  $2688  :  at  wha 
rate  must  he  sell  it  to  gain  $1.60  on  each  barrel? 

32.  A  farmer  has  a  granary  containing  449  bush.  1  pk.  2  qt. 
of  wheat ;  he  wishes  to  put  it  into  182  bags  :  how  much  must 
he  put  into  each  bag  ? 

33.  A  trader  bought  ^150  barrels  of  flour,  for  which  he  paid 
$48*15  ;  he  sold  the  same  for  $7.25  a  barrel ;  what  was  his 
profit  on  each  barrel? 

34.  How  many  sheep,  at  $1.62J  a  head,  can  be  bought  for 
^169? 

35.  How  many  canisters,  each  holding  31b.  10  oz.,  can  be 
filled  from  a  chest  of  tea  containing  58  lb.  ? 

36.  In  26  hogsheads  the  leakage  has  reduced  the  whole 
amount  to  1358  gal.  2qt.;  if  an  equal  quantity  has  leaked  out 
of  each  hogshead,  how  much  still  remains  in  each  ? 

3t.  A  man  bought  a  piece  of  land  for  $3415.25,  and  sold  it 
for  $3801.65,  by  which  transaction  he  made  $3.40  an  acre  : 
how  many  acres  were  there? 

38.  The  whole  amount  of  gold  produced  in  California  in  the 
year  1855,  was  as  follows  :  $43313281,  sent  to  the  Atlantic 
States  ;  $6500000,  sent  directly  to  England  ;  and  $8500000 
retained  in  the  country.  In  1854,  the  total  product  of  gold  iu 
California  was  $5tU5000  ;  how  much  more  was  produced  iu 
1855  than  in  1854? 

39.  If  the  forward  wheels  of  a  carriage,  are  12  feet  in  circum- 
ference, and  the  hind  wheels,  16  feet  6  inches,  how  many  more 
times  will  the  forward  wheels  turn  round  than  the  hind  wheels, 
iu  running  a  distance  of  264  miles  ? 


Ai'i'LlCATIONS.  07 

40.  If  a  certain  township  is  9  miles  long,  and  4 J  miles  wide, 
how  many  farms  of  192  acres  each  does  it  contain? 

41.  The  total  number  of  land  warrants  issued  durmg  the 
year  ending  September  30,  1855,  was  34337,  embracing  4093850 
acres  of  land :  what  was  the  average  number  of  acres  to  each 
warrant  ? 

42.  The  longitude   of  Philadelphia  is  75°  10',  and  that  o 
Kew  Orleans  89°  2',  both  west :   when  it  is  12  m.  at  Philadel- 
phia, what  is  the  time  at  New  Orleans? 

43.  The  sun  passes  the  meridian  at  12  m.,  the  moon  at  8  hr. 
30  m.  p.  M. :  what  is  the  difference  in  longitude  between  the  sun 
and  moon  ? 

44.  Two  persons,  A  and  B,  observed  an  eclipse  of  the  moon ; 
A  observed  its  commencement  at  9  hr.  42m.  p.m.;  B  was  in 
longitude  13°  20',  and  observed  its  commencement  23  minutes 
earlier  than  A  :  what  was  A^s  longitude,  and  B's  time  of  ob- 
servation ? 

45.  If  in  11  piles  of  wood  there  are  120  cords  t  cord  feet 
5  cubic  feet,  how  much  is  there  in  each  pile? 

46.  If  16cwt.  2  qr.  111b.  10  oz.  of  flour,  be  put  into  nine 
barrels,  how  much  will  each  barrel  contain? 

47.  A  miller  bought  a  quantity  of  wheat  for  $625.40,  which 
he  floured  and  put  into  barrels  at  an  expense  of  $110.12^  :  what 
profit  did  he  make  by  selling  it  for  $900? 

48.  America  was  discovered  October  11,  1492  :  how  long  to 
he  commencement  of  the  Revolution,  April  19,  1775  ? 

49.  From  a  hogshead  of  wine,  a  merchant  draws  18  bottles 
each  contaming  1  pt.  3  gi. ;  he  then  fills  three  6-gallon  demijohns, 
and  4  dozen  bottles,  each  containing  2  qt.  1  pt.  3  gi. :  how  much 
Ibmafaied  in  the  cask? 

50.  In  753689  yards,  how  many  degrees  and  statute  miles? 

51.  In  189  m.  3  fur.  6  rd.  1ft.,  how  many  feet? 


98  APF  LIGATIONS. 

62.  If  24  meu  can  build  168  rods  of  wall  in  1  day,  how  many 
rods  can  48  men  build  in  9  days  ? 

53.  A  certain  number  increased  by  1164,  and  the  sum  multi- 
plied by  209,  gives  the  j^roduct  of  1913516  :  what  is  the  number? 

54.  If  a  man  travels  146  mi.  1  fur.  14  rd.  14  ft.  in  5  days, 
how  much  is  that  for  each  one  half-day? 

55.  If  325  acres  of  land  costs  $11112.50,  how  many  acres 
can  be  bought  for  1545? 

56.  A  merchant  having  $324,  wishes  to  purchase  an  equal 
number  of  yards  of  two  kinds  of  cloth  ;  one  kmd  was  worth 
4  dollars  a  yard,  the  other  was  worth  5  dollars  a  yard :  how 
many  yards  of  each  can  he  buy? 

51.  From  one-fourth  of  a  piece  of  cloth,  containing  68  yd. 
3  qr.,  a  tailor  cut  5  suits  of  clothes :  how  much  did  each  suit 
contain  ? 

58.  A  manufacturer  having  £6  10s.,  distributed  it  among  his 
laborers,  giving  every  man  18d.,  every  woman  12d.,  and  every 
boy  lOd. ;  the  number  of  men,  women,  and  boys  was  equal : 
what  was  the  number  of  each  ? 

59.  It  is  estimated  that  1  out  of  every  1585  persons  in  Great 
Britain  is  deaf  and  dumb.  The  population,  according  to  the 
census  of  1851,  was  20936468  :  how  many  deaf  and  dumb  per- 
sons were  there  in  the  entire  population? 

60.  A  grocer,  in  packing  6  dozen  dozen  eggs,  broke  half  a 
dozen  dozen,  and  sold  the  remainder  for  1|-  cents  a  piece :  how 
much  did  he  receive  for  the  eggs? 

61.  How  much  time  will  a  man  save  in  50  years,  beginning 
with  a  leap  year,  by  rising  45  minutes  earlier  each  day? 

62.  During  the  year  1855,  there  were  shipped  to  Great  Britam 
from  the  XJnited  States,  408434  barrels  of  flour  ;  2550092  bushels 
of  wheat ;  1048540  bushels  of  corn.  Supposing  the  flour  to  have 
sold  for  $10.25  a  barrel,  the  wheat  for  $2.1 2 J  a  bushel,  and  the 
corn  for  $0.94  a  bushel,  what  was  the  value  of  the  whole  7 


APPIJCATIONS.  99 

63.  Richard  Roe  was  born  at  6  o^clock,  a.  m.,  June  24th, 
1832:  what  was  his  age  at  3  o'clock,  r.  m.,  on  the  10th  day  of 
January,  1858  ? 

64.  A  man  dying  without  making  a  will,  left  a  widow  and  4 
children.  The  law  provides,  in  such  cases,  that  the  widow  shall 
receive  one-third  of  the  personal  property,  and  that  the  remaind(5r 
shall  be  equally  divided  among  the  children.  The  estate  was 
Falued  as-  follows  :  Stocks  worth  $5000 ;  5  horses,  at  $85  each ; 
a  yoke  of  oxen,  at  $110  ;  25  cows,  at  $22  each;  150  sheep,  at 
$2  each  ;  some  lumber,  at  $45  ;  farming  utensils,  at  $174  ;  house- 
hold furniture,  at  $450  ;  grain  and  hay,  at  $380 :  what  was  the 
share  of  the  widow  and  each  child  ? 

C5.  How  many  shingles  will  it  take  to  cover  the  two  sides 
of  the  roof  of  a  building,  55  feet  long,  with  rafters  16|  feet  in 
length,  allowing  each  shingle  to  be  15  inches  long  and  4  inches 
wide,  and  to  lay  one-third  to  the  weather? 

6G.  The  longitude  of  St.  Petersburgh  is  30°  45'  east,  and  that 
of  Washington  t1°  2'  west:  what  is  the  diflference  of  longitude 
between  the  two  places ;  and  what  is  the  time  at  St.  Petersburgh 
when  it  is  6  o'clock  a.  m,  at  Washington  ? 

67.  A  vessel  sails  from  New  York  to  Liverpool.  After  a 
number  of  days,  the  captain,  by  taking  an  observation  of  the 
sun,  finds  that  his  chronometer,  which  gives  New  York  time, 
differs  1  hr.  44  m.  from  the  time,  at  the  place  of  observation.  If 
his  chronometer  shows  the  time  to  be  3  hr.  12  m.,  p.  m.,  what  is 
the  time  at  the  place  of  observation,  and  how  far  is  the  vessel 
east  of  New  York  ? 

68.  A  cistern  containing  960  gallons,  has  two  pipes  ;  45  gal- 
ons  run  in  every  hour  by  one  pipe,  and  25  gallons  run  out  by 
he  other  :  how  long  a  time  will  be  required  to  fill  the  cistern  ? 

69.  The  whole  number  of  gallons  of  rum  manufactured  in 
the  United  States  in  1850,  was  6500500  :  if  valued  at  50  cents 
a  gallon,  how  many  school-houses  could  be  built,  worth  $750 
each,  with  the  proceeds  ?  • 


JOO  APPLICATIONS. 

10.  A  speculator  sold  840  bushels  of  wheat  for  $2180,  which 
was  1^500  more  than  he  gave  for  it :  what  did  it  cost  him  a 
bushel  ? 

11.  A  farmer  sold  a  grocer  30  bushels  of  potatoes,  at  31 J 
cents  a  bushel,  for  v/hich  he  received  6  gallons  of  molasses,  at 
45  cents  a  gallon ;  60  pounds  of  mackerel,  at  6  J  cents  a  pound ; 
and  the  remainder  in  sugar,  at  10  cents  a  pound:  how  many 
pounds  of  sugar  did  he  receive  ? 

12.  If  a  man  travels  12  mi.  3  fur.  20  rd.  in  one  day,  how  long 
will  it  take  him  to  travel  114  mi.  1  fur.  at  the  same  rate? 

13.  If  a  man  sells  2  bar.  12  gal.  2  qt.  of  beer  in  one  week, 
how  much  will  he  sell  in  12  weeks? 

14.  A  hquor  merchant  had  550  pint  bottles,  400  quart  bottles, 
350  two-quart  bottles,  315  three-quart  bottles,  and  150  jugs  hold- 
ing a  gallon  each  :  how  many  barrels  of  wine  will  fill  them  ? 

15.  How  many  yards  of  carpeting,  one  yard  wide,  will  it  take 
to  cover  the  floors  of  two  parlors,  each  18  feet  long  and  16  feet 
wide ;  and  what  will  it  cost,  at  $1.33J  a  yard  ? 

16.  How  many  rolls  of  wall-paper,  each  10  yards  long  and  2 
feet  wide,  will  it  take  to  cover  the  sides  of  a  room  22  feet  long, 
16  feet  wide,  and  9  feet  high  ? 

*l*l.  Two  persons  are  1  mi.  4  fur.  20  rd.  apart,  and  are  travel- 
ling the  same  way.  The  hindmost  gains  upon  the  foremost  5  rods 
in  travelling  25  rods  :  how  far  must  he  travel  to  overtake  him  ? 

18.  A  man  sold  500  bushels  of  wheat  at  11.15  a  bushel,  and 
took  his  pay  in  sugar  at  5  cents  a  pound.  He  afterwards  sold 
one-half  of  the  sugar :  what  quantity  had  he  left  ? 

19.  A  man  bought  1  barrels  of  sugar,  at  $12.81^  a  barrel; 
he  kept  two  barrels  for  his  own  use,  and  sold  the  remainder  for 
what  the  whole  cost  him  :  what  did  he  receive  per  barrel  ? 

80.  A  flour  merchant  bought  a  quantity  of  flour  for  $18150, 
and  sold  the  same  for  $26250,  by  which  he  gained  $3  a  barrel : 
how  many  barrels  were  there  ? 


APPLICATIONS.  101 

81.  Three  men  rented  a  farm,  and  raised  964  bush.  2pk.  4qt. 
of  grain,  which  was  to  be  divided  in  proportion  to  the  rent 
paid  by  each.  The  first  was  to  have  one-half  the  whole  ;  the 
second,  one-third  the  remainder  ;  and  the  third  what  was  left : 
how  much  did  each  have? 

82.  A  vessel,  in  longitude  tO°  25' east,  sails  105°  30'  56"  west 
hen  46°  50'  east,  then  10°  5'  40"  west,  then  39°  11'  36"  east: 

*n  what  longitude  is  she  then,  and  how  many  days  will  it  take 
her  to  sail  to  longitude  17°  west,  if  she  sails  3*  20'  each  day? 

83.  A  privateer  took  a  prize  worth  $25000,  which  was  divided 
into  125  shares,  of  which  the  captain  took  12  shares;  2  lieuten- 
ants, each  5  shares ;  6  midshipmen,  each  3  shares  ;  and  the  re- 
mainder was  divided  equally  among  85  seamen :  how  much  did 
each  receive? 

84.  If  the  longitude  of  Boston  is  71°  4',  and  a  gentleman, 
in  travelling  from  Boston  to  Chicago,  finds  that  his  watch  is 
1  hr.  5  m.  44  sec.  too  fast  by  the  time  of  the  latter  place  :  what 
is  the  longitude  of  Chicago,  provided  his  watch  has  kept  accurate 
time  ? 

85.  What  time  would  it  be  in  Boston  if  it  was  8hr.  2T  m. 
30  sec,  A.  jr.,  in  Chicago  ? 

86.  What  time  would  it  be  at  Chicago  if  it  was  12  m.  at 
Boston  ? 

87.  Two  places  lie  exactly  east  and  west  of  each  other,  and 
by  observation  it  is  found  that  the  sun  comes  to  the  meridian 
of  the  latter  place  1  hour  and  16  minutes  after  the  former :  how 
far  apart  arc  they  in  minutes  and  degrees  of  longitude? 

88.  In  12  bales  of  cloth,  each  bale  containing  16  pieces,  and 
each  piece  containing  20  ells  English,  how  many  yards  ? 

89.  A  speculator  gave  $8968  for  a  certain  number  of  barrels 
of  flour,  and  sold  a  part  of  it  for  $2618,  at  $7  a  baiTcl,  and 
by  so  doing  lost  $2^  on  each  barrel :  for  how  much  must  he  sell 
the  remainder,  to  gain  $1060  on  the  whole? 


102  APPLICATIONS. 

90.  How  many  eagles  can  be  made  from  24  lb.  4  oz.  6  pwt. 
18  gr.  of  gold,  making  no  allowance  for  waste,  if  each  eagle 
weighs  11  pwt.  9gr.  ? 

91.  A  man  paid  $3284.82  for  some  wheat.  He  sold  T40 
bushels  at  2  dollars  a  bushel ;  the  remainder  stood  him  in  $1.42 
a  bushel :  how  many  bushels  did  he  purchase  ? 

92.  A  man  sold  105  A.  2  R.  20  P.  of  land  for  as  many  dollars 
as  there  were  perches  of  land,  payable  in  instalments,  at  the  rate 
of  1  dollar  an  hour.  If  the  contract  was  closed  at  12  o'clock,  m., 
April  1st,  1856,  what  length  of  time  will  be  allowed  the  pur- 
chaser to  pay  the  debt,  reckoning  365  days  6  hours  to  the 
year  ? 

98.  The  sum  of  2  numbers  is  98,  and  their  difference  is  46 : 
what  are  the  numbers? 

94.  A  farmer  paid  $76  dollars  more  for  a  horse  than  for  a 
cow  J  he  paid  $190  for  both:  what  was  the  value  of  each? 

95.  How  many  days  intervene  between  March  5th  and  August 
21st,  both  days  inclusive? 

96.  A  merchant  buys  810  barrels  of  flour,  at  $9.50  a  barrel. 
He  finds  one-half  of  it  injured,  and  is  willing  to  lose  one- 
quarter  on  the  value  of  that  part :  how  much  loss  was  that 
on  each  half  barrel? 

9t.  Three  merchants.  A,  B,  and  C,  are  engaged  together  in 
business,  and  gain  in  one  year  $24612.  This  amount  is  to  be 
equally  divided  among  them,  after  paying  A  $6t5,  and  B  $812, 
for  extra  services.    How  much  did  each  receive  ? 

98.  Four  merchants  are  in  partnership.  Their  apparent  profits 
during  the  year  amount  to  $56895  ;  but  they  have  expended  for 
clerk  hire,  $6750  ;  for  rent,  $3500  ;  for  insurance,  $156 ;  and  for 
incidental  expenses,  $364.  The  first  is  to  have  $250  for  extra 
services  ;  the  second,  $175  for  travelling  expenses  ;  and  the  third, 
$95  for  various  articles  furnished  by  him  to  the  concern.  What 
was  th^  share  of  profit  of  each,  after  paying  these  expenses  ? 


PROPERTIES  OF   NUMBERS.  103 

PROPERTIES    OF    NUMBERS. 
£jzact  Divisors — Prime  Numbers. 

104.  An  Exact  Divisor  of  a  number,  is  any  number,  except  1 
ind  the  number  itself,  that  will  divide  it  without  a  remainder. 

105.  One  Number  is  divisible  by  another,  when  the  remainder 
is  0. 

106.  An  Odd  Number  is  one  not  divisible  by  2. 

107.  An  Even  Number  is  one  divisible  by  2. 

108.  A  Prime  Number  is  one  which  has  no  exa<it  divisor. 

.    1,  2,  3,  5,  7,  11,  13,  It,  19,  23,  &c., 
are  prime  numbers. 

109.    What  numbers  are  esact  divisors. 

1.  Any  Factor  of  a  composite  number  is  an  exact  divisor 
of  the  number. 

2.  Three  is  an  exact  divisor  of  any  number,  the  sum  of  whoso 
digits  is  divisible  by  3. 

3.  Four  is  an  exact  divisor  of  a  number  when  it  will  exactly 
divide  the  number  expressed  by  the  ivfo  right-hand  digits. 

4.  Five  is  an  exact  divisor  of  every  number  whose  right-hand 
figure  is  0  or  5. 

5.  Six  is  an  exact  divisor  of  any  even  number  of  which  3  is 
an  exact  divisor. 

101  What  is  an  exact  divisor  ? — 105.  When  is  one  number  divisible 
by  another? — lOG.  What  is  an  odd  number? — 107.  What  is  an  even 
niunber? — 108.  Wliat  is  a  prime  number? 

109. — 1.  Is  a  factor  of  a  composite  number  an  exact  divisor? 

2.  What  numbers  will  3  exactly  divide?  3.  What  numbers  -will  4 
exactly  divide?  4.  Whht  numbers  will  5  exactly  divide?  5.  What 
numbers  will  G  exactly  di^-idc?  6.  Vv'liat  numbers  v.'ill  9  exactly  dl- 
vide?     7.  What  numbers  will  10  exactly  divide? 


104  PROPERTIES   OF   NUMBERS. 

6.  Nine  is  an  exact  divisor  of  any  number,  the  sum  of  wliose 
digits  is  divisible  by  9. 

1.  Ten  is  an  exact  divisor  of  every  number  whose  right-hand 
figure  is  0. 

110.    To  find  the  prime  factors  of  a  composite  number. 

1.  What  are  the  prime  factors  of  18  ? 

Analysis. — Every  factor    of   a    composite  j^g  _  q  x^  2 

number  is  an  exact  divisor.     If    any  divisor  =  3  y  3  y  2 

is  a  composite  number,  resolve  it  again  into 
factors,  till  each  shall  be  prime:  hence, 

Uvery  composite  onimher  is  equal  to  the  product  of  all  its 
prime  factors :  hence,  it  is  divisible  hy  each  of  them. 

Thus,         24  =  3X    8  =  3X2X4  =  3X2X2X2; 
and  60  =  5x  12  =  5x2x6  =  5x2x3x2. 

2.  What  are  the  prime  factors  of  the  composite  number  105? 
Analysis. — Tliree  being   an   exact  divisor,  and  a  3)105 

prime  number,  we  divide  by  it,  giving  the  quotient  b\'^b 

35;  then,  5  and   7  are   prime  divisors  of  the  quo-  -— 

tient:  hence,  3,  5,  and  7  are  the  prime  factors  of  105. 

Hence,  to  find  the  prime  factors  of  any  composite  number, 
H/Tile. — Divide  the  given  number  by  any  prime  number  that 
is  an  exact  divisor :  then  divide  the  quotient  by  any  other  exact 
prime  divisor,  and  so  on,  till  a  quotient  is  found  which  is  a 
prime  number:  the  several  divisors  and  the  last  quotient  will 
he  the  prime  factors. 

Examples. 

1.  What  are  the  prime  factors  of  9?  10?  12?  14?  16? 
18?   24?   27?   28? 

2.  What  are  the  prime  factors  of  30?  22?  32?  36?  38? 
40?   45?   49? 

110.  What  is  the  rule  for  finding  the  prime  factors  of  a  composite 
number  ? 


PROPERTIES  OF  NUMBKliS.  105 

3.  What  are  the  prime  factors  of  50?  56?  58?  60?  64? 
66?   68?    70?    72? 

4.  What  are  the  prime  factors  of  76?  78?  80?  82?  84? 
86?   88?    90? 

5.  What  are  the  prime  factors  of  100?  102?  104?  275? 
•^SO?    472?    160?   836? 

6.  What  are  the  prime  factors  of  105?  106?  108?  110? 
il5?    116?    120?    125?    1125?   360? 

Note. — llio  prime  factors,  when  the  numbers  are  small,  may  gen- 
erally be  seen  by  inspection.  The  teacher  can  easily  increase  the  num- 
ber of  examples. 

111.  To  find  the  prime  factors  common  to  two  or  more  com- 
posite nimibers. 

1.   What  are  the  common  prime  factors  of  90,  120,  and  150  ? 

Analysis. — It  is  jjlain  that  3  is  an  exact  2  ^  90         120         150 

divisor  of  all  the  numbers  :  hence,  it  is  a  ^ 

prime  factor  of  them.     Since  3  is  an  exact  '^ )  4o   .   .     bU   .   .      to 

divisor  of  the  quotients,  it  is  a  prime  factor  5)15..     20   .   .     25 

of  them  ;  and  since  o  will  divide  the  second  o              a               5 
set  of  quotients,  it  is  a  prime  factor.     The 

quotients  3,  4,  and  5  have  no  exact  divisor;  therefore,  2,  3,  and  5  are 
all  the  common  prime  factors :   hence, 

The  common  prime  factors  of  two  or  more  numbers,  are 
the  exact  divisors  common  to  them  all. 

Examples. 

1.  What  are  the  prime  factors  common  to  150,  210,  and  270  ? 

2.  What  are  the  prime  factors  common  to  42,  126,  and  168  ? 

3.  What  are  the  prime  factors  common  to  105,  315,  and  525  ? 

4.  What  are  the  prime  factors  common  to  84,  126,  and  210  ? 

5.  What  are  the  prime  factors  common  to  168,  256,  410, 
and  820  ? 

111.  What  is  the  rule  for  finding  the  primo  factors  common  to  two 
or  more  comtiosite  numbers? 


106  CANCELLATION. 

CANCELLATION. 

112.   Cancellation  is   a  method  of  shortening  Arithmetical 
operations  by  omitting  or  cancelling  common  factors. 

1.   Divide  36  by  18.     First,  36^:9x4;   and  18^=9x2. 

Analysis. — Thirty-six  divided  by  18  is  equal  operation. 

to    9  X  4    divided   by   9X2:    by  cancelling,  or        35        0x4 
striking  out  the  9's,  we  have  4  divided  by  2,        Jg  ~  0^x2  ~ 
which  is  equal  to  2. 

Note. — The  figures  cancelled  are  slightly  crossed. 

113.     Principles— Operations— and  Rule. 
The  operations,  in  cancellation,  depend  on  two  principles  : 

1.  The  cancelling  of  a  factor,  in  any  number,  is  equivalent  to  divid- 
ing the  number  by  that  factor. 

2.  If  the  dividend  and  divisor  be  both  di\ided  by  the  same  number, 
the  quotient  will  not  be  chang<id. 

1.  Divide  56  by  32. 

^         ,  ,  ,.     .  ,        ,  ,  OPKRATION. 

Analysis. — Resolve    the    dividend   and  An        n 

divisor  into  factors,  and  then  cancel  those         —  z=z =z  -  =  l^. 

which  are  common.  ^^        ^^^        ^ 

2.  In  12  times  25,  how  many  times  45  ? 

Analysis. — We  see  that  9  is  a  factor  of  72  and  operation. 

45.     Divide  by  9,  and  write  the  quotient  8  over  72,  8        5 

and  the  quotient  5  below  45.     Again,  5  is  a  factor  /T^J  X  ^^  _ 

of  25  and  5.     Divide  25  by  5,  and  write, the  quo-  ^^ 

tient  5  over  25.     Dividing  5  by  5,  reduces  the  di-  ^ 
visor  to  1 :   hence,  the  true  quotient  is  40. 

Rule. 

I.  Resolve  the  dividend  and  divisor  into  //zeiVj;ri??ie /actors 
or  conceive  them  to  be  so  resolved : 

II.  Cancel  the  common  faclors,  and  then  divide  the  product 

112.  What  is  Cancellation? 

113.  On  what  principles  does  the  operation  of  cancellation  depend? 
What  is  the  rule  for  the  operation? 


CANCKLLATION.  107 

of  the  remaining  factors  of  the  dividend  by  the  product  of  the 
remaining  factors  of  the  divisor. 

Notes. — 1.  Since  every  factor  ia  cancelled  by  division,  the  quotient  1 
always  takes  the  place  of  the  cancelled  factor. 

2.  If  one  of  the  numbers  contains  a  factor  equal  to  the  product  of 
two  or  more  factors  of  the  other,  all  such  factors  may  bo  cancelled. 

3.  If  the  product  of  two  or  more  factors  of  the  dividend  is  equal  to 
the  product  of  two  or  more  factors  of  the  divisor,  such  factors  may  bo 
cancelled. 

Examples. 

1.  What  is  the  quotient  of  2x4x8x13x1x1 6,  divided  by 
26x14x8? 

2.  What  is  the  quotient  of  42  x  3  x  25  x  12,  divided  by 
28x4x15x6? 

3.  What  is  the  quotient  of  125x60x24x42,  divided  by 
25x120x36x5? 

4.  How  many  tunes  ia  11x39x1x2  contamed  in  44  x 
18x26x14? 

5.  What  is  the  quotient  of  8  tunes  240  multiplied  by  6  times 
114,  divided  by  24  times  5T  multiplied  by  6  times  15? 

6.  What  is  the  value  of  (22  +  8  +  16)  x  (18  +  10  +  21)  divided 
by  (9+5  +  7)  x(15+8)? 

7.  Divide  (140  +  86-34)  X  (107-19)  by  (237  -  141)  x 
(17  +  20-15)? 

8.  Divide  [12x5-2  x  9]  x  (42  +  30)  by  (5x8)x(2x9)x 
(lO  +  n)? 

9.  What  is  the  quotient  of  240x441x16  divided  by  175  x 
56x27? 

10.  What  is  the  quotient  of  64  times  840  multiplied  by  9 
times  124,  divided  by  32  times  560  multiplied  by  4  times  31  ? 

11.  How  many  dozen  of  eggs,  worth  14  cents  a  dozen,  mus*t 
be  given  for  18  pounds  of  sugar,  worth  7  cents  a  pound? 

12.  A  dairyman  sold  5  chcescf?,  each  weighing  40  pounds,  at 
9  cents  a  pound  :  how  many  pounds  of  tea,  wortli  50  cents  a 
f^bfind,  must  lie  receive  for  the  clieescs? 


108  CANCELLATION. 

13.  Bought  12  yards  of  cloth,  at  $1.84  a  yard,  and  paid  for 
it  in  potatoes  at  48  cents  a  bushel :  how  many  bushels  of  pota- 
toes will  pay  for  the  cloth? 

14.  How  many  firkins  of  butter,  each  containing  56  pounds,  at 
25  cents  a  pound,  will  pay  for  4  barrels  of  sugar,  each  weighing 
115  pounds,  at  8  cents  a  pound? 

15.  A  man  bought  10  cords  of  wood,  at  20  shillings  a  cord, 
and  paid  in  labor  at  1 2  shillings  a  day  :  how  many  days  did 
he  labor? 

16.  How  many  pieces  of  cloth,  each  containing  36  yards,  at 
$3.50  a  yard,  must  be  given  for  96  barrels  of  flour,  at  $10.50 
a  barrel  ? 

IT.  A  farmer  exchanged  492  bushels  of  wheat,  worth  $1.84 
n  bushel,  for  an  equal  number  of  bushels  of  barley,  at  8*J  cents 
a  bushel ;  of  corn,  at  60  cents  a  bushel ;  and  of  oats,  at  45  cents 
a  bushel :   how  many  bushels  of  each  did  he  receive  ? 

18.  How  many  barrels  of  flour,  worth  $t  a  barrel,  must  be 
given  for  250  bushels  of  oats,  at  42  cents  a  bushel? 

19.  If  48  acres,  of  land  produce  2484  bushels  of  corn,  how. 
many  bushels  will  120  acres  produce? 

20.  A  man  worked  12  days, 'at  9  shillings  a  day,  and  re- 
ceived in  payment  wheat  at  16  shillings  a  bushel  :  how  many 
bushels  did  he  receive? 

21.  A  grocer  sold  6  hams,  each  weighing  14  pounds,  at  10 
cents  a  pound,  and  received  in  payment  apples,  at  48  cents  a 
bushel  :  how  many  bushels  of  apples  did  he  receive  ? 

22.  How  long  will  it  take  a  man,  travelling  36  miles  a  day, 
to  go  the  same  distance  that  another  man  travelled  in  15  days, 
at  the  rate  of  2t  miles  a  day  ? 

23.  A  man  took  four  loads  of  apples  to  market,  each  load 
containing  12  barrels,  and  each  barrel  3  bushels.  He  sold  them 
at  45  cents  a  bushel,  and  received  in  payment  a  number  of 
boxes  of  tea,  each  box  containing  20  pounds,  worth  72  cents 
a  pound  :  how  many  boxes  of  tea  did  he  receive  ? 


COMMON   MULTIPLE.  109 


LEAST    COMMON    MULTIPLE. 

114.  A  Multiple  of  a  number  is  any  product  of  which  the 
number  is  a  factor ;  hence,  any  multiple  of  a  number  is  exactly 
divisible  by  the  number  itself. 

115.  A  Common  Multiple  of  two  or  more  numbers  is  any 
number  which  each  will  divide  without  a  remainder. 

116.  The  Least  Common  Multiple  of  two  or  more  numbers 
is  the  least  number  which  they  will  separately  divide  without  a 
remainder. 

117.    Principles — Operations — and  Rule. 

1.  Any  divisible  number,  is  divisible  by  any  prime  factor  of  the  exact 
divisor. 

2.  If  a  number  has  several  exact  divisors,  it  will  be  divisible  by  all 
their  prime  factors. 

3.  Hence,  the  question  of  finding  the  least  common  multiple  of  scv 
eral  numbers  is  reduced  to  finding  a  number  which  shall  contain  all 
their  prime  factors,  and  none  others. 

1.  "What  is  the  least  common  multiple  of  6,  12,  and  18? 

ANAiiYSis. — Having  placed  the  given  num-  operation. 

bers  in  a  line,  if  we  divide  by  2,  we  find  the  2  )  6   .   .  1 2   .   .  1 8 

quotients  3,  C  and  9 ;   hence,  2  is  a  prime  o  \  q  g  q 

factor  of  all  the  numbers.     Dividing  by  3, 

we  find  that  3  is  a  prime  factor  of  the  quo-  i   .  .     ^   .   . 

ticnts  3,  G,  and  9 ;  and  hence,  the  quotients       2x3x2x3  =  36 
2  and  3  are    prime   factors  of  12  and  18 ; 

therefore,  the  prime  factors  of  all  the  numbers  are  2,  3,  2  and  3 ;  and 
their  product,  3G,  is  the  least  common  multiple. 

114.  What  is  a  multiple  of  a  number? — 115.  What  is  a  common 
nmltiple.of  two  or  more  numbers?  — 110.  What  is  the  least  common 
multiple  of  two  or  more  numbers? 

117.  What  is  the  first  principle  on  which  the  operation  for  finding 
the  hmst  common  multiple  depends  ?  What  is  the  second  ?  What  is 
the  tldrd?    Give  the  rule  for  finding  the  least  common  multiple. 


110  COMMON   MULTIPLE. 

Rule. 

I.  Place  the  numbers  on  the  same  line,  and  divide  by  any 
PRIME  number  that  will  exactly  divide  two  or  more  of  them, 
and  set  down,  in  a  line  beloxo,  the  quotients  and  the  undivided 
numbers: 

II.  Then  divide  as  before,  until  there  is  no  prime  number 
jr cater  than  1  that  will  exactly  divide  any  two  of  them : 

III.  Then  multiply  together  the  divisors  and  the  numbers  of 
the  lower  line,  and  their  product  will  be  the  least  common 
multiple. 

Note. — If  tlie  numbers  have  no  common  prime  factor,  tlieir  product 
will  be  tlieir  least  common  multiple. 

Examples. 

1.  What  is  the  least  common  multiple  of  4,  9,  10,  15,  18, 
20,  21? 

2.  What  is  the  least  common  multiple  of  8,  9,  10,  12,  25, 
32,  75,  80? 

3.  What  is  the  least  common  multiple  of  1,  2,  3,  4,  5,  6, 
7,  9? 

4.  What  is  the  least  common  multiple  of  9,  16,  42,  63,  21,. 
14,  T2? 

5.  What  is  the  least  common  multiple  of  1,  15,  21,  28,  35, 
100,  125? 

6.  What  is  the  least  common  multiple  of  15,  16,  18,  20,  24, 
25,  27,  30? 

7.  What  is  the  least  common  multiple  of  9,  18,  27,  36,  45,  54  ? 

8.  What  is  the  least  common  multiple  of  4,  10,  14,  15,  21  ? 

9.  What  is  the  least  common  multiple  of  7,  14,  16,  21,  24  ? 

10.  What  is  the  least  common  multiple  of  49,  14,  84,  168,  98  ? 

11.  A  can  dig  9  rods  of  ditch  in  a  day;  B,  12  rods  in  a 
day ;  and  C,  1 6  rods  in  a  day :  Avhat  is  the  Smallest  number 
of  rods  that  would  afford  exact  days  of  labor  to  each,  working 
alone  ?     lu  what  time  would  each  do  the  whole  work  ? 


GREATEST  COMMON   DIVISOR.  Ill 

12.  A  blacksmith  employed  4  classes  of  workmen,  at  $15, 
$16,  $21,  and  $24  per  month,  for  each  man  respectively,  paying 
to  each  class  the  same  amount  of  wages.  Required  the  least 
amount  that  will  pay  either  class  for  1  mouth ;  also,  the  number 
of  men  in  each  class? 

13.  A  farmer  has  a  uum])er  of  bags  containing  2  biisliels 
each  ;  of  barrels,  containing  3  bushels  each  ;  of  boxes,  containiug 
7  bushels  each;  and  of  hogsheads,  containing  15  bushels  each: 
what  is  the  smallest  quantity  of  wheat  that  would  fill  each  an 
exact  number  of  times,  and  how  many  times  would  that  quan- 
tity fill  each  ? 

14.  Four  persons  start  from  the  same  point  to  travel  round 
a  circuit  of  300  miles  in  circumference.  A  goes  15  miles  a  day; 
B,  20  milcvS ;  C,  25  miles  ;  and  D,  30  miles  a  day.  How  many 
days  must  they  travel  before  they  will  all  come  together  again  at 
the  same  point,  and  how  many  times  will  each  have  gone  round  ? 

Note. — First  find  the  number  of  days  that  it  will  take  each  to  travel 
round  the  circuit, 

GREATEST    COMMON    DIVISOR. 

118.  A  Common  DiyisoR  of  two  or  more  numbers,  is  any 
number  that  will  divide  each  of  them  without  a  remamder ;  hence, 
it  is  always  a  common  factor  of  the  numbers. 

119.  The  Greatest  Common  Divisor  of  two  or  more  numbers, 
is  the  greatest  number  that  will  divide  each  of  them  without  a 
remainder ;  hence,  it  is  their  greatest  common  factor. 

120.  Two  numbers  are  said  to  be  prime  to  each  other,  whon 
they  have  no  common  divisor. 

Note. — Since  1  will  divide  evci'y  number,  it  is  not  reckoned  anidOfr 
the  common  divisors. 

118.  Wluit  is  a  common  divisor? — 119.  What  la  the  grcat<'at  «i)TU 
nion  divisor? 

120.  When  are  two  numbers  said  to  be  vriric  to  cnoh  oMicr? 


112  GREATEST  COMMON   DIVISOR. 

121.  To  find  the  greatest  common  divisor  of  two  or  more 
numbers,  when  the  numbers  are  small. 

Since  an  exact  divisor  is  a  factor,  the  greatest  common  divi- 
sor of  the  given  numbers  will  be  theu*  greatest  common  factor : 
hence, 

I.  Resolve  each  number  into  its  prime  factors,  and  observe 
those  which  are  common  to  all  the  numbers: 

II.  Multiphj  the  common  factors  together,  and  their  produci 
will  be  the  greatest  common  divisor. 

Examples. 

1.  What  is  the  greatest  common  divisor  of  12  and  20? 

Analysis.— There  are  three  prime  factors  operation. 

in  12 ;  viz.,  3,  3,  and  3 :  there  are  three  prime  -.  o  _  o       9       9 

factors  in  30;  viz.,  3,  3,  and  5.     The  factors  1^  —  J  X  J  X  o. 

3  and  3  are  common ;  hence,  3X3  =  4  is  20  =  2x2x5. 
tlie  greatest  common  divisor. 

2.  What  is  the  greatest  common  divisor  of  18  and  36? 

3.  What  is  the  greatest  common  divisor  of  12,  24,  and  60  ? 

4.  What  is  the  greatest  common  divisqr  of  15,  50,  and  40  ? 

5.  What  is  the  greatest  common  divisor  of  24,  18,  and  144  ? 

6.  What  is  the  greatest  common  divisor  of  50,  100,  and  80  ? 

7.  What  is  the  greatest  common  divisor  of  56,  84,  and  140  ? 

8.  What  is  the  greatest  common  divisor  of  84,  154,  and  210  ? 

122.  To  find  the  greatest  common  divisor,  when  the  numbers 
are  large. 

This  method  depends  on  the  foUowmg  principles : 

1.   Any  number  which  will    exactly    divide  illusteation. 

wo  numbers   separately,  will  divide  their  dif-  oq 0 00 

erenee;    else,  we  should  have  a  whole  number 
equal  to  a  fraction^  which  is  impossible. 

131.  How  do  you  find  the  greatest  common  divisor,  when  the  num- 
bers ar6  small  ? — 133.  On  what  princii^le  does  finding  the  greatest  com- 
mon divisor  depend?    Give  the  rule. 


GREATEST  COMMON   DIVISOR.  118 

2,  Any  number  that  will  exactly  divide  the  difference  of  two  num- 
bers, and  one  of  them,  will  exactly  divide  the  other :  else,  we  should 
have  k  whole  number  equal  to  a  fraction,  which  is  impossible. 

3.  Any  number  which  will  exactly  divide  another,  will  divide  any 
multiple  of  that  other ;  because,  the  first  dividend  which  is  divisible 
is  a  factor  of  the  multiple. 

1 .  Let  it  be  required  to  find  the  greatest  common  divisor  of 
the  numbers  216  and  408. 

,.    .  OPERATION. 

Analysis. — The  greatest  common  divi- 
sor cannot  be  greater  than  the  least  num-         2iio)  4U(5  [i 
ber,  216.     Now,  as  216  wiU  divide  itself,  .^ 

let  us  see  if  it  will  divide  408;,  for,  if  it  192)  216  (1 

will,  it  is  the  greatest  common  divisor.  192 

Making  the  division,  we  find  a  quotient  1,  91^  1 09  /'R 

and  a  remainder,  192 ;  hence,  216  is  not  a  *"       192 

common  divisor. 

The  greatest  common  divisor  of  216  and  408  will  divide  the  remain- 
der 192 ;  and  if  192  will  exactly  divide  216,  it  will  be  the  greatest  com- 
mon divisor.  We  find  that  192  is  contained  in  216  once,  and  a  remain- 
der 24.  The  greatest  common  divisor  of  192  and  216  will  di\'ide  the 
remainder  24;  and  if  24  will  exactly  divide  192,  it  will  also  divide  216, 
and  consequently  408;  now,  24  exactly  divides  192,  and  hence  is  the 
greatest  common  divisor  sought. 

Rule. 

Divide  the  greater  number  by  the  less,  and  then  divide  the 
preceding  divisor  by  the  remainder,  and  so  on,  till  nothing  re- 
mains :  the  last  divisor  mill  be  the  greatest  common  divisor. 

Principles  from  the  Rule. 

1.  If  the  last  remainder  is  1,  the  numbers  are  pnme  to  eacli  ot/ier. 

2.  If,  in  the  course  of  the  operation,  any  one  of  the  remainders  is 
a  'prime  number,  and  will  not  exactly  divide  the  preceding  divisor,  it 
is  certain  that  there  is  no  common  divisor. 

3.  To  find  the  greatest  common  divisor  of  three  or  more  numbers, 
find  the  greatest  common  divisor  of  two  of  them,  and  then  the  divisor 
of  this  common  divisor,  and  of  the  tliird  number,  and  go  on. 


114  GREATEST  COMMON  DIVISOR. 

Examples. 

1.  What  is  the  greatest  common  divisor  of  3328  and  4592? 

2.  What  is  the  greatest  common  divisor  of  2205  and  4501  ? 

3.  What  is  the  greatest  number  that  will  divide  16082  and 

tst'ior 

4.  What  is  the  greatest  number  that  will  divide  620,  1116,  ^ 
«ud  1488  ? 

5.  What  is  the  greatest  common  divisor  of  5270,  5952,  5394, 
«nd  3038  ? 

6.  What  is  the  greatest  common  divisor  of  461t,  1695,  6642, 
and  8424? 

I.  A  farmer  has  315  bushels  of  corn,  and  810  bushels  of 
vfheat ;  he  wishes  to  draw  the  corn  and  wheat  to  market  sepa- 
rately in  the  fewest  number  of  equal  loads  :  how  many  bushels 
must  he  draw  at  a  load  ? 

8.  The  Illinois  Central  Railroad  Company  have  15t50  acres 
of  land  in  one  location,  and  21*725  acres  in  another.  They  wish 
to  divide  the  whole  into  lots  of  equal  extent,  containing  the 
greatest  number  of  acres  that  will  give  an  exact  division :  how 
many  acres  will  there  be  in  each  lot  ? 

9.  A  man  has  a  corner  lot  of  land,  1044  feet  long  and  t44 
feet  wide.  The  adjacent  sides  are  bounded  by  the  highway,  and 
he  wishes  to  build  a  board  fence  with  the  fewest  panels  of  equal 
length :  what  must  be  the  length  of  the  panels  ? 

10.  A  farmer  has  231  bushels  of  barley,  369  bushels  of  oats, 
and  393  bushels  of  wheat,  all  of  which  he  wishes  to  put  into 
the  smallest  number  of  bags  of  equal  size,  without  mixing : 
how  many  bushels  must  each  bag  contain  ? 

II.  Three  persons,  A,  B,  and  C,  ngree  to  purchase  a  lot  of 
63  cows  at  the  same  price  per  head,  provided  each  man  can  thus 
mvest  his  whole  money.  A  has  $286 ;  B,  $462  ;  and  C,  $638 : 
how  many  cows  could  each  man  purcha.<?G  ? 


COMMON  FRACTIONS. 


115 


COMMON    FRACTIONS. 

123.  An  Integral,  or  whole  number,  is  the  unit  1,  or  a 
collection  of  such  units. 

Note. — All  integral  numbers  are  formed  by  the  continual  addition 
of  1 :  as,  1  H-  1  =  2,    3  +  1  =  3,  &c. 

124.  A  Unit  is  a  single  thing;  as,  an  apple,  a  chair,  a  hat, 
&c. ;  and  is  denoted  by  1. 

If  a  unit  be  divided  into  two  equal  parts,  each  part  is 
called,  one-half. 

If  a  unit  be  divided  into  three  equal  parts,  each  part  is 
called,  one-third. 

If  a  unit  be  divided  into  four  equal  parts,  each  part  is 
called,  one-fourth. 

If  a  unit  be  divided  into  twelve  equal  parts,  each  part  is 
called,  one-twelfth;  and  if  it  be  divided  into  any  number  of 
equal  parts,  we  have  a  like  expression  for  each  part. 

The  parts  are  thus  written  : 


J 

is  read. 

one-half. 

i 

, 

one-third. 

i 

. 

one-fourth 

i 

, 

one-fifth. 

i 

. 

one-sixth. 

¥ 

1 
To 


is  read. 


one-seventh. 

one-eighth. 

one-tenth. 

one-fifteenth. 

one-fiftieth. 


The  -J,  is  an  entire  haf ;  the  \,  an  entire  third;  the  J,  an 
entire  fourth  ;  and  the  same  for  each  of  the  other  equal  parts ; 
hence,  each  equal  X)art  is  an  entire  thing,  and  is  called  a  frac- 
tional unit. 


123.  What  is  an  integral,  or  whole  number?  IIow  are  integral 
uunibcrs  formed? 

124.  Wliat  is  a  unit?  By  what  is  it  denoted?  What  is  each  part 
called  when  the  unit  1  is  divided  into  two  equal  parts?  When  it  ia 
divided  into  three?    Into  four?    Into  five?    Into  twelve? 


116  OOMMON   FRACTIONS. 

125.  The  Unit  of  a  Fraction  is  the  single  thing  that  is 
divided  into  equal  parts 

126.  The  Fractional  Unit  is  one  of  the  equal  parts  of  the 
unit  that  is  divided. 

127.  A  Fraction  is  a  fractional  unit,  or  a  collection  of 
such  units. 

Note. — In  every  fraction,  let  the  pupil  distinguish  carefully  between 
tlie  tmit  of  the  fraction  and  the  fractional  unit.  The  first  is  the  whole 
thing  from  which  the  fraction  is  derived ;  the  second,  one  of  the  equal 
parts  into  which  that  tiling  is  divided. 

128.  Every  whole  number,  except  1,  has  a  fractional  unit 
corresponding  to  it :   thus  the  numbers, 

2,     3,     4,     5,     6,     t,     8,     9,     10,     &c., 

have,  corresponding  to  them,  the  fractional  units, 

h    h    h    h    h    h    h    h    to,    &c. 

129.    Expressing   Fractions. 

Each  fractional  unit  may,  hke  the  unit  1,  become  the  base 
of  a  collection:  thus,  suppose  it  were  requked  to  express  2  of 
each  of  the  fractional  units,  we  should  then  write 

J  which  is  read  2  halves  =  J  x  2. 
f  ...  2  thirds  =  J  X  2. 
f  .  .  .  2  fourths  =1x2. 
f  .  .  .  2  fifths  =  i  X  2. 
&c.,  &c.,  &c. 

If  it  were  required  to  express  3  of  each  of  the  fractional 
units,  we  should  write 

f  which  is  read  3  halves  =  ^  x  3. 
f  .  .  .  3  thirds  =  i  x  3. 
I  ...  3  fourths  =  i  X  3. 
f  ....  3  fifths  =  i  X  3. 
&c.,  &c.,  &c. 


COMMON   FRACTIONS.  117 

Hence,  if  we  suppose  a  second  unit  to  be  divided  into  tho 
same  number  of  equal  parts,  such  parts  may  be  expressed  in 
the  same  collection  with  the  parts  of  the  first:  thus, 

I  is  read,       3  halves. 

J  .        .         T  fourths. 

y  .         .16  fifths. 

J^  .         .18  sixths. 

^  .        .25  sevenths. 

A  whole  number  may  be  expressed  fractionally  by  writing  1 
below  it  for  a  denominator.    Thus, 

3  may  be  written  f  and  is  read,  3  ones. 

6  ....  f      ...  5  ones. 

6  ....  f      ...  6  ones. 

8  ....  f       ...  8  ones. 

But  3  ones  are  equal  to  3,  5  ones  to  5,  6  ones  to  6,  and 
8  ones  to  8  ;  hence,  the  value  of  a  number  is  not  changed  by 
placing  1  under  it  for  a  denominator:    Hence,  we  see, 

1.  That  Fractions  are  expressed  by  two  numbers,  one  written  above 
the  other,  with  a  line  between  them. 

2.  That  every  fraction  may  be  divided  into  two  factors,  one  of 
which  is  tho  fractional  unit,  and  the  other  the  number  denoting  how 
many  times  the  fractional  unit  is  taken. 

130.  The  Denominator  is  the  number  written  below  the  line. 
It  shows  into  how  many  equal  parts  the  unit  of  the  fraction 
is  divided. 

125.  What  is  the  unit  of  a  fraction  ?— 12G.  What  is  the  fractional 
tmit  ? — 127.  What  is  a  fraction  ? — 128.  What  fractional  unit  correspondij 
to  the  whole  number  5?    What  to  6?    What  to  14? 

129.  May  a  fractional  unit  become  the  base  of  a  collection  ?  How 
are  fractions  expressed?  Into  how  many  factors  may  every  fraction 
bo  divided?    Wliat  are  they? 

130.  What  is  the  denominator  of  a  fraction  ? 


118  COMMON    FRACTIONS. 

131.  The  Kuiierator  is  the  number  written  above  tlie  line. 
It  shows  how  many  fractional  units  are  taken. 

132.  The  Terms  of  a  fraction  are  the  numerator  and  de- 
nominator taken  together :  hence,  every  fraction  has  two  terms. 

133.  The  Yalue  of  a  fraction  is  the  quotient  of  the  numer 
at  or  divided  by  the  denominator. 

134.  The  Analysis  of  a  fraction  is  the  naming  of  its  unit — 
its  fractional  unit — and  the  number  of  fractional  units  taken. 

135.    Analysis  of  Fractions. 

How  is  the  fraction  |  to  be  interpreted  ? 

1.  The  unit  of  the  fraction  is  1. 

2.  The  unit  of  the  fraction  is  divided  into  8  equal  parts ;  hence,  the 
fractional  unit  is  one-eighth. 

3.  Seven  fractional  units  are  taken.  In  the  fraction  |,  the  base  of 
the  collection  of  fractional  units  is  ^,  but  it  is  not  the  pnmary  base. 
For,  I  is  one-eighth  of  the  unit  1 ;  hence,  the  primary  base  of  every 
fraction  is  the  unit  1. 

The  expression  may  also  be  interpreted  as  the  quotient  of  t 
divided  by  8.  In  the  latter  case,  the  thing  divided  is  the  num- 
ber 7 ;  in  the  former,  it  was  the  number  1.  The  value  in  both 
cases  is  the  same ;  for,  seven  times  one-eighth  of  1,  is  equal  to 
one  of  the  8  equals  of  1.  Hence,  a  fractional  expression  has 
the  same  form  as  an  unexecuted  division. 

136.    Principles  and  Properties   of  Fractions. 

1.  A  fraction  is  a  fractional  unit,  or  a  collection  of  such  units. 

2.  The  denominator  sliows  into  how  many  equal  parts  the  unit  of 
the  fraction  is  divided. 

131.  Wliat  is  the  numerator  of  a  fraction  ? — 132.  Wliat  are  the  terms 
of  a  fraction?  How  many  terms  has  every  fraction ? — 133.  What  is  tlie 
value  of  a  fraction  ? — 134.  What  is  the  analysis  of  a  fraction  ? 

135.  Analyze  the  fraction  -}.  What  is  the  base  of  the  collection? 
What  is  the  primary  base?    What  else  does  |  express? 

13G.   Explain  tlie  principles  and  properties  of  Fractions. 


COMMON     KliACriONS.  119 

8.   Tho  numerator  showa  how  many  fractional  units  are  taken, 

4.  The  value  of  every  fraction  is  ei^ual  to  tho  numerator  divided  by 
the  denominator. 

5.  Wlien  the  numerator  is  lees  than  the  denominator,  the  value  of 
the  fraction  is  less  than  1. 

G.  When  the  numerator  is  equal  to  the  denominator,  the  value  of  the 
firaction  is  equal  to  1. 

7.  When  the  numerator  is  greater  than  the  denominator,  the  value 
of  the  fraction  is  greater  than  1. 

137.    Writing  and  Reading  Fractions. 

1.  Read  and  analyze  the  following  fractions  : 

S»        tV'         h        TSf        V»        ^f        Toi' 

2.  Write  15  of  the  19  equal  parts  of  1.  Also,  3t  of  the  49 
equal  parts  of  1.     Write  24-thirtieths. 

3.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit 
one-fortieth,  express  2t  fractional  units.  Also,  95.  Also,  106. 
Also,  87.    Also,  41. 

4.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit 
one-68th,  express  45  fractional  units.  Also,  56.  Also,  85.  Also, 
95.     Also,  37. 

5.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit  one- 
90th,  express  9  fractional  units.    Also,  87.    Also,  16.    Also,  65. 

138.    Six  Kinds  of  Fractions. 

1.  A  Proper  Fraction  is  one  whose  numerator  is  less  than 
the  denominator. 

The  following  are  proper  fractions : 

2»       i»       h       if      y>       ¥>       To>       9>       t* 


137.  Give  an  example  in  writing,  reading,  and  analyzing  fractions 

138.  How  many  kinds  of  fractions  are  there?    Name  and  describe 
each. 


120  COMMON   FRACTIONS. 

2.  An  Improper  Fraction  is  one*  whose  numerator  is  equal 
to,  or  exceeds  the  denominator/  The  following  are  improper 
fractions : 

3.  A  6  8L  9.  J_2  IJL  1_0 

2'  3>  5>  7»  8»  6»  7»  7* 

Note. — Such  a  fraction  is  called  improper,  because  its  value  equals 
or  exceeds  1. 

3.  A  Simple  Fraction  is  one  whose  numerator  and  denomina 
tor  are  both  whole  numbers.    The  following  are  simple  fractions : 

i35.8.98.6X  ^ 

4>  2>  6»  7'  2>  3>  3>  6* 

Note. — A  simple  fraction  may  be  either  proper  or  improper. 

4.  A  Compound  Fraction  is  a  fraction  of  a  fraction,  or 
several  fractions  connected  by  the  word  of.  The  followmg  are 
compound  fractions : 

1  of  i,       4  of  1  of  4,       1  of  3,       ^  of  i  of  4. 

5.  A  Mixed  Number  is  made  up  of  a  whole  number  and  a 
fraction.     The  following  arc  mixed  numbers: 

3|,         41,         Ql         6f,         ^,         31 

6.  A  Complex  Fraction  is  one  which  has  a  fraction  in  one 
or  both  of  its  terms.     The  following  are  complex  fractions; 

(i)        A       ii)         IM 

FUNDAMENTAL   PEINCIPLES. 

139.    Let  it  be  required  to  multiply  f  by  4. 

Analysis.— In  f  there  are  3  fractional  operation. 

units,  each  of  which  is  ^,  and  these  are  to         |.  x  4  =  -'^-  =  ^. 
be  taken  4  times.    But  three  things  taken 

4  times,  give  12  things  of  tbe  same  kind;  that  is,  12  eiglitlis ;  hence, 
the  product  is  4  times  as  great  as  the  multiplicand;  tlierefore. 

Proposition  I. — If  the  numerator  of  a  fraction  he  multiplied 
by  any  number,  the  value  of  the  fraction  will  be  multiplied 
as  many  times  as  there  are  units  in  the  multiplier. 


COMMON   FRACTIONS.  121 


Examples. 


1.  Multiply  I    by  G,  by  1. 

2.  Multiply  I   by  4,  by  9. 

3.  Multiply  /y  by  11,  by  12. 


6.  Multiply  ^  by  3,  by  4. 
6.  Multiply  lA  by  7,  by  9. 
t.  Multiply  41  by  5,  by  10. 


4.  Multiply  ^j  by  12,  by  14.     |     8.  Multiply  §J  by  3,  by  11. 
140.    Let  it  be  required  to  multiply  yj  by  4. 


Analysis. — In  y'^  there  are  5  fractional  operation. 

5      V   1   —        5 


units,  each  of  which  is  j\.    If  we  divide  s    v>  >i  —      s —  __ 


the  denominator  by  4,  the  quotient  is  3, 
and  the  fractional  unit  becomes  ^,  which  is  4  times  as  great  as  ^^; 
because,  if  ^  be  divided  into  4  equal  parts,  each  part  will  be  ^j.  If 
we  take  this  fractional  unit  5  times,  the  result,  |,  will  be  4  times  as 
great  as  j\;  therefore, 

Proposition  II. — If  the  denominator  of  a  fraction  he  divided 
by  any  number^  the  value  of  the  fraction  will  be  midliplied  as 
many  times  as  there  are  unUs  in  the  divisor. 

Hence,  to  multiply  a  fraction  by  any  number,  divide  its  de- 
nominator. 

Examples. 


1.  Multiply  U  by  8,  by  4,  by  2. 

2.  Multiply  ^  by  2,  3,  4,  6,  8. 

3.  Multiply  3^  by  6,  5,  10,  15. 

4.  Multiply  i|  by  2,  3,  4,  6,  8. 


5.  Multiply  A  by  4,  5,  10,  20. 

6.  Multiply  ^V  by  1,  by  5. 

Y.  Multiply  f^  by  21,  6, 1,  3,  2. 

8.  Multiply  if  by  3,  4,  6,  9,  12. 


141.     Let  it  be  required  to  divide  ^j  by  3. 

Analysis. — In  y\  there  are  9  fractional  operation. 

units,  each  of  which  is  yV»  ^^^  thesQ  are         _9  -i.  3  —  9±1  —   s 
to  be  divided  by  3.    But  9  tilings,  divided 


139.  What  is  proved  in  Proposition  I.? — 140.  What  is  proved  in 
PropoBition  II.? 


122 


COinfON    FRACTIONS. 


by  3,  gives  3  things  of  the  same  Jchid  for  a  quotient ;  hence,  the  quo- 
tient is  3  elevenths,  a  number  which  is  one-third  of  yy;  hence, 

Proposition  III. — If  the  numerator  of  a  fraction  he  divided 
hy  any  number,  the  value  of  the  fraction  will  he  divided  into 
as  many  equal  parts  as  there  are  units  in  the  divisor. 

Examples. 


1.  Divide  i|  by  2,  4,  8,  16. 

2.  Divide  -fA  by  2,  1,  and  14. 

3.  Divide  f  §  by  2,  5,  4,  and  10. 

4.  Divide  f  f  by  5,  6,  10,  15,  20. 


5.  Divide  if  by  2,  3,  6,  and  9. 

6.  Divide  f  ^  by  3,  6,  8,  12. 

7.  Divide  f|  by  3,  9,  and  21. 

8.  Divide  f|  by  6,  9,  21,  54. 


142.    Let  it  be  required  to  divide  jj  by  3. 


9 

TT 


3  = 


Analysis. — In  y\  there  are  9  fractional  orEUATioN. 

units,  each  of  which  is  y^-.  Now,  if  we 
multiply  the  denominator  by  8,  it  becomes 
83,  and  the  fractional  unit  becomes  ^'j,  which  is  one-third  part  of 
y\.  If,  then,  we  take  this  fractional  unit  9  times,  the  result,  ^^3,  is 
just  one-third  part  of  ^j;  hence,  we  have  divided  the  fraction  y^ 
by  3 :   therefore,  we  have 

Proposition  TV. — If  the  denominator  of  a  fraction  he  multi- 
plied hy  any  numher,  the  value  of  the  fraction  will  he  divided 
into  as  many  equal  parts  as  there  are  units  in  the  multiplier. 

Hence,  to  divide  a  fraction,  multiply  the  denominator. 


1.  Divide  f  by  6,  T,  and  8. 

2.  Divide  f  by  5,  4,  and  9. 

3.  Divide  if  by  3,  4,  and  12. 

4.  Divide  ff  by  6,  8,  and  11. 


Examples. 

5.  Divide  |f  by  1,  5,  and  3. 

6.  Divide  if  by  1,  8,  and  6. 

7.  Divide  f  |  by  3,  1,  and  11. 

8.  Divide  fj  by  8,  4,  and  10 


141.  What  is  proved  in  Proposition  III.? — 143.  What  is  proved  iu 
Proposition  IV.  ? 


COMMON     FRACTIONS.  123 


143.     Multiply  both  terms  of  the  fraction  f  by  4. 

Analysis. — In  f,  the  fractional  unit  is  |,  and  it         operation. 
is  taken  3  times.     Bj  multiplying  the  denominator       3x4      12 
by  4,  the  fractional  unit  becomes  ^V*  *h®  value  of       5  X  4  "^  20 
which    is    one-fourth    of  |.     By    multiplying  the 
D  jmerator  by  4,  we  increase  the  number  of  fractional  units  taken, 
4  times ;    that  is,  we  increase   the  number  of  parts  taJcen  just   as 
mamj   times   as  wo  diminish   the  value  of  each  part;  hence  the 
value  of  the  fraction  is  not  changed:  therefore, 

Proposition  Y. — If  both  terms  of  a  fraction  be  multiplied  by 
the  same  number,  the  value  of  the  fraction  will  not  be  changed. 

Examples. 

1.  Multiply  both  terms  of  the  fraction  J  by  4,  by  6,  and  by  5 

2.  Multiply  both  terms  of  ^\  by  5,  by  8,  by  9,  and  11. 

3.  Multiply  both  terms  of  jf  by  1,  by  8,  and  9. 

4.  Multiply  both  terms  of  ^  by  5,  8,  6,  and  12. 
6.   Multiply  both  terms  of  f  |  by  2,  3,  4,  and  5. 

144.    Divide  both  terms  of  the  fraction  j^  by  3. 

Analysis. — In  yj,  the  fractional  unit  is  ^j,  and  operation. 

it  is  taken  0  times.    By  dividing  the  denominator         6-^3       2 
by  3,  the   fractional  unit  becomes  |,  the  value  of        15-7-3       5* 
which  is  3  times  as  great  a^  yj.    By  dividing  the 
numerator  by  3,  we   diminish  the  number  of  fractional  units  taken 
3  times ;  that  is,  we  diminish  the  number  of  parts  taken  just  as  many 
times  as  we  increase  the  value  of  the  fractional  unit:  hence,  the  value 
of  the  fraction  is  not  changed ;  therefore. 

Proposition  YI. — If  both  terms  of  a  fraction  be  divided  by  the 
he  same  number^  the  value  of  the  fraction  will  not  be  changed. 


143.  What  is  proved  in  Proposition  V.?— 144.  What  is  proved  la 
Proposition  VI.? 


124  REDUCTION  OP 


Examples. 

1 .  Divide  both  terms  of  f  by  2  and  by  4. 

2.  Divide  both  terms  of  f  by  3. 

3.  Divide  both  terms  of  ||  by  2,  3,  4,  6,  and  12. 

4.  Divide  both  terms  of  ff  by  2,  4,  8,  and  16. 

5.  Divide  both  terms  of  ^  by  2,  3,  4,  6,  and  12. 

6.  Divide  both  terms  of  ^  by  2,  3,  4,  6,  and  36. 


REDUCTION    OF    FRACTIONS. 

145.  Reduction  of  Fractions  is  the  operation  of  chaLging 
a  fractional  number  from  one  unit  to  another  without  altering 
its  value. 

146.  The  lowest  terms  of  a  fraction  are  when  the  numerator 
and  denominator  are  prime  to  each  other. 

CASE     I. 

147.  To  reduce  a  whole  number  to  a  fraction  having  a  given 
denominator. 

I.  Reduce  17  to  a  fraction  whose  denominator  shall  be  5, 

Analysis. — To  reduce  17  to  such  a  fraction  is  operation. 

the  same  as  to  reduce  17  to  fifths.     In  17  there       17  X  5  =85 
are  17  times  as  many  fifths  as  there  are  in  1.     In  17  —  _8_5 

1    there  are  5  fifths;   therefore,  in  17  there  are 
17  times  5  fifths,  or,   85  fifths;  hence, 

Rnle. — Multiply  the  whole  number  by  the  denominator,  and 
ivrite  the  product  over  the  required  denominator, 

145.  What  is  reduction  of  fractions  ? 

146.  Wliat  are  the  lowest  terms  of  a  fraction  ? 

147.  How  do  you  reduce  a  whole  number  to  a  fraction  having  a 
given  denominator? 


COMMON   FRACTIONS.  125 


Examples. 

1.  Change  18  to  a  fraction  whose  denominator  shall  be  *l. 

2.  Change  25  to  a  fraction  whose  denominator  shall  be  12. 

3.  Change  19  to  a  fraction  whose  denominator  shall  be  8. 

4.  Change  29  to  a  fraction  whose  denominator  shall  be  14. 

5.  Change  65  to  a  fraction  whose  denominator  shall  be  37. 

6.  Reduce  145  to  a  fraction  having   9  for  its  denominator. 

7.  Reduce  450  to  twelfths. 

8.  Reduce  327  to  a  fraction  having  36  for  its  denominator. 

9.  Reduce  97  to  a  fraction  having  128  for  its  denominator. 

10.  Reduce  167  to  eighty-ninths. 

11.  Reduce  325  to  a  fraction  whose  denominator  shall  be  75. 

CASE      II. 

148.  To  reduce  a  mixed  number  to  an  equivalent  improper 
fraction. 

1.   Reduce  12^  to  its  equivalent  improper  fraction. 

Analysis. — Since    in   any  num-  operation. 

ber  there  are  7  times  as  many  7ths  12x7  =  84  sevenths, 

as  units  1,  there  will  be  84  sev-  add                       5  sevenths, 

enths  in  12 :     To  these  add  5  sev-  gives      12f-  =  89  sevenths, 

enths,  and  the   equivalent  fraction  ^^^^  _  sj^ 
becomes  89  sevenths.     Hence, 

Rule. — Multiply  the  whole  number  by  the  denominator: 
to  the  iwoduct  add  the  numerator,  and  place  the  sum  over  the 
given  denominator. 

Examples. 

1.  Reduce  39|-  to  its  equivalent  improper  fraction. 

2.  Reduce  112j®o  to  its  equivalent  improper  fraction. 

148.  How  do  you  reduce  a  mixed  number  to  an  equivalent  improper 
fraction  ?  • 


120  REDUCTION  OF 

3.  Reduce  42TJi  to  its  equivalent  improper  fraction. 

4.  Reduce  67 Off  to  an  improper  fraction. 

5.  Reduce  367 jf  4-  to  an  improper  fraction. 

6.  Reduce  Silj-fj  to  an  improper  fraction. 

7.  Reduce  67426fff  to  an  improper  fraction. 

8.  How  many  200ths  in  6751|J? 

9.  How  many  151ths  in  187 ^Vt  ? 

10.  Reduce  149f  to  an  improper  fraction. 

11.  Reduce  375ff  to  an  improper  fraction. 

12.  Reduce  1*74949 1|-|9^  to  an  improper  fraction. 

13.  Reduce  4834|-|-  to  an  improper  fraction. 

14.  Reduce  1789f  to  an  improper  fraction. 

15.  In  125|-  yards,  how  many  sevenths  of  a  yard  ? 

16.  In  375J  feet,  how  many  fourths  of  a  foot? 

17.  In  464if  hogsheads,  how  many  sixty-thirds. 

18.  In  96JJq  acres,  how  many  640ths  of  an  acre? 

19.  In  984jY2  pounds,  how  many  112ths  of  a  pound? 

20.  In  353^0-  years,  how  many  366ths  of  a  year  ? 

21.  How   many   one   hundred   and   thirty-fifths   are  there  in 
the  mixed  number  87y3j? 

22.  Place  4  sevens  in  such  a  manner  that  they  shall  express 
the  number  78. 

23.  By  means  of  5  threes  write  a  number  that  is   equal  to 
334. 

CASE     III. 

149.    To  reduce  an  improper  fraction  to  an  equivalent  whole 
or  mixed  number. 

1.   In  ^-p-  how  many  entire  units  ? 


149.  How  do  you  reduce  an  improper  fraction  to    an    equivalent 
mixed  number? 


COMMON   FliACTIONS.  127 

Analysis. — Since    there  are  5  fifths    in  1   unit,  operation. 

there  will  be,  in  278  fiftlis,  as  many  units  1  as  5  is  5)2t8 

contained  times  in  278,  viz.,  55  and  ^  times.     Hence,  55f  • 
the  following 

Rule. — Divide  the  numerator  by  the  denominator^  and  the 
(Quotient  will  he  the  equivalent  ivhole  or  mixed  number. 

Examples. 
Reduce  the  following  fractions  to  whole,  or  mixed  nnmbers. 


1.  Reduce  J^. 

2.  Reduce  ^^^-. 

3.  Reduce  VaV- 

4.  Reduce  i|?g^. 

6.  Reduce  Jy'/-  pounds. 

6.  Reduce  — Jf-  days, 

t.  Reduce  -^^  yards. 

8.  Reduce  4f^. 


9.  Reduce  ^Wtr^  acres. 

10.  Reduce  Vir- 

11.  Reduce  ^^^^^. 

12.  Reduce  ^M|^. 

13.  Reduce  ^^^. 

14.  Reduce  VVr- 

15.  Reduce  ^\%%^. 
10.   Reduce  ^J^. 


CASE     IV. 
150.    To  reduce  a  fraction  to  its  lowest  terms. 

1.   Reduce  -^^^  to  its  lowest  terms. 

Analysis. — By  inspection,  it  is  seen  that  1st  operation. 

5  is  a  common  factor  of  the  numerator  and  ^)tW  ^^  sT* 

denominator.     Dividing  by  it  we  have  ^j. 
We  then  sec   that  7  is  a  common  factor  of  */35  —  s- 

14  and  35 :  dividing  by  it,  we  have  f.    Now, 

2   and  5    are  prime  to  each   other;    therefore,  the  fraction  f  is  in 
its  lowest  terim. 

2d.   The  greatest  common  divisor  of  70  and  175,  is  35  (Art.  119); 
if  we  divide  both  terras  of  tfie  fraction  by 
it,  we   obtain  |.     The  value  of  the  fraction  2d  operation. 

is  not  changed  in  either  operation,  since  the  35)  7  o   _.  2^ 

nnraGrator    and    denominator    are    both    di- 
vided by  tlic  same  number  (Art.  144) :  hence, 


128 


REDUCTION  OF 


Rule. 

I.  Divide  the  numerator  and  denominator ^  successively,  by 
all  their  common  factors:  Or, 

II.  Divide  the  numerator  and  denominator  by  their  greatest 
common  divisor. 

Examples. 
Keduce  the  following  fractions  to  their  lowest  terms  : 

12.  Keduce  ^%*q. 
U.  Reduce  m. 


1.  Reduce  -^q. 

2.  Reduce  ^^y. 

3.  Reduce  JJf 
i 
5 


Reduce  iff|. 


14. 

Reduce 

t'AV 

15. 

Reduce 

8343 

9T4y 

16. 

Reduce 

T^- 

n. 

Reduce 

an- 

18. 

Reduce 

T%V 

19. 

Reduce 

mil 

20. 

Reduce 

5%V5- 

21. 

Reduce 

sffjo- 

22. 

Reduce 

.^"o- 

Reduce  f^. 

6.  Reduce  f|J. 

7.  Reduce  t^tV.. 

8.  Reduce  f  f  by  2d  method. 

9.  Reduce  fjf       *' 

10.  Reduce  ^2^     " 

11.  Reduce  yVA     " 


CASEY. 
151.    To  reduce  a  compound  fraction  to  a  simple  fraction. 
1.   What  is  the  equivalent  fraction  of  f  of  f  ? 
Analysis. — Three-fifths  of  4  is  three  times  operation. 

I  of  ^ :   1  fifth  of  I  is   3^   (Art.  142) ;  and  3         3x4       12 
times  a^j  is  ^f  (Art.  139) ;  hence,  §  of  |  =  if ; 
hence. 


6  X  T  -  35 


Rnle. 

I.  If  there  are  mixed  numbers,  reduce  them  to  improper 
fractions  : 

II.  When   there  are  common   factors  in    the    numerators 

and  de7iominators,  cancel  them: 


150.  How  do  you  reduce  a  fraction  to  its  lowest  terms?     151.  How 
do  you  reduce  a  compound  fraction  to  a  simple  one? 


COMMON   FRACTIONS.  129 

III.   Multiply  the  numerators  together  for  a  new  mtmer- 
ator,  and  the  denominators  together  for  a  new  denominator. 

Examples. 

1.  Reduce  f  of  J  of  f  to  a  simple  fraction. 

2.  Reduce  f  of  J  of  J  to  a  simple  fraction. 
3    Roduce  J  of  |  of  2J  to  a  simple  fraction. 

4.  Change  |  of  |  of  f  of  3 J  to  a  simple  fraction. 

5.  Change  ^q  of  f  of  J  of  j\  to  a  simple  fraction. 
G.    What  is  the  value  of  |  of  J  of  J  of  12|  ? 

7.  What  is  the  value  of  f  of  |  of  4J  ? 

8.  What  is  the  value  of  j%  of  ^  of  5y\  ? 

9.  Reduce  |-  of  9J  of  6y  of  2  J  to  a  whole  or  mixed  number. 

10.  Reduce  ^j  of  -Jj  of  21|  to  a  whole  or  mixed  number. 

11.  Reduce  J  of  f  of  |  of  j^j^  of  /j  to  a  simple  fraction. 

12.  Reduce  ^^  of  j'g  of  ^Yt  of  y  to  a  simple  fraction. 

13.  Reduce  3|-  of  ^  of  ^^j  of  49  to  a  simple  fraction. 

CASE    VI. 

152.     To   reduce  fractions  of  different  denominators  to  eqaiv 
alent  fractions   that  shall  have   a  common   denominator. 

1.  Reduce  f,  |,  and  f  to  a  common  denominator. 

Analysis.— Multiplying  both  terms  operation. 

of  the  first  fraction  by  20,  the  pro-       2x5x4  =  40  1st  num. 
duct    of    the    other    denominators,       4  X  3  X  4  =  48  2d  num. 
gives   f °.     Multiplying    both    terms       3  X  5  X  3  =  45  3d  num. 
of  tlie  second  fraction    by  12,  the       3  X  5  X  4  =  60  deuom. 
product  of  the  other  denominators, 

gives  jf.  Multiplying  both  terms  of  the  third  by  15,  the  product 
of  the  other  denominators,  gives  |^.  In  each  case,  both  terms  of 
the  fraction  have  been  multiplied  by  the  same  number;  therefore, 
the  value  is  not  changed  (Art.  143) :   hence, 

Rule. — I.   Reduce  mixed  niimhers  to   improper  fractions, 
and  compound  to  simple  fractions,  when  necessary 


130  KEDUCTION   OF 

TT.  Multiply  the  7nimerator  of  each  fraction  by  all  the  de- 
nominators except  its  own,  for  the  new  numerators,  and  all 
the  denominators  together  for  a  common  denominator. 

Note — Wlien  tlie  numbers  are  small,  the  work  may  be  performed 
mentally ;  thus, 

112   Vromp    2  0     L5     24  .    n,.pi    2    1    2  become   i^    2.0    so 


Examples. 
Reduce  the  followhipr  fractions  to  common  denominators  : 


1.  Reduce  f,  5|,  and  f. 

2.  Reduce  f ,  f,  \,  and  \  of  5. 

3.  Reduce  ^,  4i,  2f,  and  |. 

4.  Reduce  f,  |-,  |,  \,  and  2  J. 

5.  Reduce  ^  of  3,  f ,  f ,  and  f . 

6.  Reduce  2i  of  3i    and  4|. 


7.  Reduce  f  of  J,  and  6f. 

8.  Reduce  4f,  2J,  5^,  and  6. 

9.  Reduce  5},  f ,  3^,  and  3|. 

10.  Reduce  J  of  5|,  and  4f 

11.  Reduce  4i  of  31    and  ^. 

12.  Reduce  61  of  2,  f ,  5f ,  and  J. 


Note. — We  may  often  shorten  the  work  of  multiplying  the  nimier- 
ator  and  denominator  of  each  fraction  by  such  a  number  as  will 
make  the  denominators  the  same  in  all. 

Reduce  the  following  fractions  to  common  denominators. 

1.  Reduce  f,  /j;  h  ^'^^  f  to  a  common  denominator. 

2.  Reduce  f,  ^j,  and  -J  to  a  common  denominator. 

3.  Reduce  4 J,  ^q,  and  7 J  to  a  common  denominator. 

4.  Reduce  lOf,  |,  and  7|  to  a  common  denominator. 

5.  Reduce  6 J,  J,  and  7|  to  a  common  denominator. 

6.  Reduce  f,  |-,   141,  and  3 J  to  a  common  denominator. 

7.  Reduce  j'^,  f,  2f,  and  If  to  a  common  denominator. 

8.  Reduce  f,  i   if,  and  J  to  a  common  denominator. 
■9.  Reduce  -fj,  f,  ^,  and  ^  to  a  common  denominator. 
10.  Reduce  21,  51,  j%,  and  4y^2  to  a  common  denominator. 

152.  How  do  you  reduce  fractions  to  a  common  denominator? 


COMMON    FKAGTIONS.  131 

CASE     VII. 
153.    To    reduce  fractions    to    their  least   common  denominator. 

The  least  common  denominator  is  the  least  common  multiple 
of  the  denominators. 

I.  Reduce  f,  f,  and  J,  to  their  least  common  denominator. 

Analysis. — If  there  are  mixed  numbers  or  compound  fractions, 
thej  must  be  reduced.  We  then  find  the  least  common  multiple 
of  the  denominators  4,  6,  and  9,  which  is  36.  This  number  is 
divided  by  each  denominator,  to  ascertain  by  what  the  terms  of 
the  fraction  must  be  multiplied  to  reduce  it  to  86ths. 


OPERATION. 


2)4 


3)2     .  .     3     .  .     9 
2     .  .     1     .  .     3 


LELiST  COMMON  DENOMINATOR. 

2x3x2x3  =  36. 


(36 

(36 
(36 


4)  X  3  =  2t  1st  numerator. 
6)  X  5  =  30  2d  numerator. 
9)     X     4     =     16     3d  numerator. 


Therefore,  the  fractions,  reduced  to  their  least  common  denom- 
inator, arc 

U,   ^l   a^^  if. 

Rule. 

I.  Find  the  least  common  multixile  of  the  denominators: 
this  will  he  the  least  common  denominator  of  the  fractions : 

II.  Divide  the  least  common  denominator  by  the  denomina- 
tor of  each  fraction^  separately;  multiply  the  quotient  by  the 
numerator  and  place  the  product  over  the  least  common  de- 
nominator ;  the  results  icill  be  the  neio  and  equivalent  frac- 
tions. 

153.  How  do  you  reduce  fractions  to  their  least  common  dcnoniina' 
tor? 


132  REDUCTION  OF 

Examples. 

1.  Keduce  |,  j,  and  y\  to  their  least  common  denominator. 

2.  Reduce  y^,  f,  and  if  to  their  least  common  denominator. 

3.  Reduce  2f ,  -f^,  and  ^  to  their  least  common  denominator. 

4.  Reduce  5|,  43-^2*  ^^^  24  ^^  *^^^^  ^^^^^  common  denominator. 
6.  Reduce  8j^3-,  -|,  and  g'^g-  to  their  least  common  denominator 
6.  Reduce  9j  5-,  2^,  and  /g-  to  their  least  common  denominator 
T.  Reduce  2J,  S^\,  and  -^-^  to  their  least  common  denominator 

8.  Reduce  3^^,  |-,  ^,  and  y%  to  their  least  common  denominator. 

9.  Reduce  f ,  ^y,  and  5^  to  their  least  common  denominator. 

10.  Reduce  4y^,  12^,  and  -^^  to  their  least  common  denominator. 

11.  Reduce  6f,  Bj^q,  and  2^^^  to  their  least  common  denominator. 

12.  Reduce  ^^y,  2^,  and  1^  to  their  least  common  denominator. 

13.  Reduce  6 J,  6^,  g^g,  and  -^  to  their  least  denominator. 

DENOMINATE    FRACTIONS. 

154.  A  Denominate  Fraction  is  one  whose  unit  is  denominate. 
Thus,  f  of  a  yard  is  a  denominate  fraction. 

CASE     VIII. 

155.  To  change  a  denominate  fraction  from  a  greater   unit  to 
a  less. 

1.  In  J  of  a  yard,  how  many  inches  ?  operation. 

Analysis. — Since  3  feet  make  a  yard,  1r 

I  yd.  =  I  of  ^  feet;   and  since  12  inches  _>/_><_—  — 

make  one  foot,   I  yard  ==  |  of  f  of  V  |        ^         ^  ^ 

inches  =  ^  =  Zll  inches.  ~  .SU 

Rule. 

Multiply  the  fraction  hy  the  units  of  the  scale,  in  succession, 
till  you  reach  the  required  unit. 

154.  Wliat  is  a  denominate  fraction? — 155.  How  do  you  change  a 
denounnate  fraction  from  a  greater  to  a  less  unit  ? 


COMMON  FRACTIONS.  133 


CASE     IX. 

156.  To  change  a  denominate  fraction  from  a  less  unit  to  a 
greater. 

1.  Reduce  I  of  a  pound  to  a  fraction  of  a  ton. 

Analysis. — Since  one  pound  is  ^\  of  a  quarter,  |  lb.  =  ^  of  o'-  qr.; 
?.iul  since  one  quarter  is  |  of  a  cwt.,  f  lb.  =f  of  ^s  of  I  cwt;  and 
Bince  one  cwt.  is  -^^  of  a  ton, 

9  ^^-  =  9  ^^  25  ^^  4  ^^  ^  =  4500  ^'^ 

5 

Rule. — Divide  the  fraction,  that  is,  multiply  the  denomina- 
tor by  the  units  of  the  scale,  in  succession,  till  the  required 
unit  is  reached. 

CASE     X. 

157.  To  find  the  value  of  a  denominate  fraction  in  integers  ol 
lower   denominations. 

1.    What   is    the  value  of  I  lb.  opekation. 

Troy  ?  1 

12 
Analysis. — I  lb.  =  f  of  V"  =  V  =  9)84 

91  oz. :  i  oz.  =  f  of  V  =  V  =  n        oz    9       3 
pwt.  :   f  pwt.  =  I  of  2-4  =  144  ^  16  2^ 

gr. :  hence,  9JgO 

Rule. — 3fuUiply  the  numerator  W^-  ^  •  -^ 

of  the  fraction  by  the  units  of  the 
scale,  and  divide  the  product  by  the 
denominator  ;  if  there  is  a  remain- 
der, treat  it  in  the  same  way,  till 
the  Inquired  denomination  is  reached.  Hie  quotients  of  the 
several  operations  will  form  the  answer. 


24 

9)144 

grTlG 

Ans.  9  oz.,  6  pwt.,  16  gr 


15G  How  do  you  cliango  a  denominate  fraction  from  a  less  te  a 
greater  imit? — 157.  How  do  you  find  tlie  value  of  a  denominate  frac- 
tion in  integers  of  lower  denominations. 


134  REDUCTION   OF 

Examples. 

1.  Reduce  £^  to  the  fraction  of  a  farthing, 

2.  Reduce  f  ton  to  the  fraction  of  a  pound. 

3.  Reduce  ^  week  to  the  fraction  of  a  minute 

4.  Reduce  /g  lb.  Troy  to  the  fraction  of  a  grain. 

5.  Reduce  |  inch  to  the  fraction  of  a  rod. 

6.  Reduce  f  inch  to  the  fraction  of  a  yard. 

7.  Reduce  ij  of  a  second  to  the  fraction  of  a  degree. 

8.  Reduce  ij  of  a  cubic  foot  to  the  fraction  of  a  cord. 

9.  What  is  the  value  of  £^^  ?   of  £-^^  ? 

10.  Find  the  value  of  J  mile  :    the  value  of  f  mile. 

11.  What  is  the  value  of  |-  furlong? 

12.  Reduce  f  penny  to  the  fraction  of  a  guinea. 

13.  Reduce  ^  farthing  to  the  fraction  of  6  guineas. 

14.  Reduce  j^j  hour  to  the  fraction  of  5  seconds. 

CASE     XI. 

158.  To  reduce  a  compound  denominate  number  to  a  fraction 
of  a  given  denomination. 

1.   Reduce  3oz.  14pwt.  15  gr.  to  the  fraction  of  a  pound. 

Analysis.  —  3oz.  14pwt.  15gr.  =  1791gr.  In  lib.  there  are 
iJTCOgr. ;    therefore   3oz.  14pwt.  15 gr.  is  i^lJlb. 

Rule. — Reduce  the  compound  number  and  the  unit  of  the 
given  denomination  to  the  lowed  unit  named  in  either,  and 
then  divide  the  first  result  by  the  second. 

Examples. 

1.  Change  7  fur.  28  rd.  2  yd.  to  the  fraction  of  a  mile. 

2.  Reduce  17s.  6d.  2  far.  to  the  fraction  of  a  £. 

3.  Reduce  lOcwt.  oqr.  161b.  to  the  fraction  of  a  ton. 

158.  How  do  you  reduce  a  compound  number  to  a  fraction  of  a 
given  donomiualiou? 


COMMON   FRACTIONS.  135 

4.  Reduce  9oz.  5|pwt.  to  the  fraction  of  1  lb.  Troy. 

6.  Reduce  5  da.  161ir.  40  m.  to  the  fraction  of  a  week. 

6.  Change  3pk.  Tqt.  Ipt.  to  the  fraction  of  a  bushel. 

T.  Change  3qr.  3na.  linch  to  the  fraction  of  1  yard. 

8.  Change  18s.  8d.  3  far.  to  the  fraction  of  £1  9s.  6d. 

9.  Change  Js.  to  the  fraction  of  £|. 
10.  Change  ijd.  to  the  fraction  of  £^. 


ADDITION. 

159.  Addition  of  Fractions  is  the  operation  of  finding  the 
sum  of  two  or  more  fractions. 

160.  The  sum  of  two  or  more  fractions  is  a  number  which 
contains  tlie  same  fractional  unit  as  many  times  as  it  is  con- 
tained in  all  the  fractions  taken  together. 

CASE     I. 
161.    When  the  fractions  have  the  same  unit. 

1.  What  is  the  sum  of  J,  f,  |,  and  §  ? 

Analysis. — In  this  example  the  unit 
of  the  fraction  is  1,  and  the  fractional 
unit  ^.  There  is  1  half  in  the  first,  3 
halves  in  the  second,  6  in  the  third, 
and  3  in  the  fourth ;  hence,  there  are  13  halves  in  all,  equal  to  6|. 

2.  What  is  the  sum  of  £^  and  je§  ? 

Analysis. — The  unit  of  loth  frac-  operation. 

tions  is  £1.     In  tlie  first,  the  fractional  £\  =  £\ 

unit   is   £^,  and    in    the    second,    £.}.  j£|  =  £^ 

These  fractional  units,  being  different,  jgs  ^  jgj.  _ .  ^i_  _  £\\ 
cannot  be  expressed  in  one  collection. 

But  £i  =  £,\   and  £\  =  £|,   in    each  of  which    expressions    tlie 
fractional  unit  is  £\ :  hence,  their  sum  is  £J  =  £1|. 

150.  What  is  Addition  of  Fractions?— 1  GO.  What  is  the  sum  of  two 
or  more  fractions? — 161.  How  do  you  add  fractions  which  have  the 
Btuuo  unit? 


opekation. 

1+3+6+3 

=  13 

hence,  J^  =  6^ 

sum. 

136  ADDITION  OP 

Rule. 

1.  W/ie?i  the  fractions  have  the  same  denominator^  add 
their  numerators^  and  place  the  sum  over  the  common  de- 
nominator : 

II.  When  they  have  not  the  same  denominator^  reduce 
compound  fractions  to  simple  ones,  and  then  reduce  cdl 
to  a  common  denominator,  and  add  as  before. 

Note. — 1.  Reduce  each  fraction  to  its  lowest  terms  before  adding. 

2.  After  the  addition  is  performed,  reduce  every  result  to  its 
simplest  form ;  that  is,  improper  fractions  to  mixed  numbers,  and  the 
fractional  parts  to  their  lowest  terms. 

162.     When  each  of  two  fractions  has  1  for  a  numerator. 

1.  What  is  the  sum  of  \  and  -J-? 

Analysis. — Reducing    to    a  common  operation. 

denominator,  we  find  the  fractions  to       11  ^    \^  ■'^ 

bo  -^^  nnd  ^^,  and  their  sum  to  be  ^f.       5       7       35      35      35' 
That  is,  tlie  sum  of  two  fractions  whose 

numerators  are  each  1,  is  equal  to   the  sum  of  their  denominators 
divided  ly  their  product. 

2.  What  is  the  sum  of  i  and  J?  of  ^  and  J  ?  of  |  and  J? 
of  J  and^V? 

3.  What  is  the  sum  of  ^  and  -^q  ?  of  Jj  and  j^g  ?  of  J  and 
J  ?  of  i  and  \  ? 

163.  When  there  are  mixed  numbers. 
1.  What  is  the  sum  of  12f,  llf,  and  15f-? 

OPERATION. 

WhoU  IN'um'bera.  Fractions. 

12  +  11  +  15=38  f +f +f-TV5+A'5 +T¥5=fSf =li8l : 

then,  38  +  lfgf=39i§f.  Ans. 

162.  What  is  the  sum  of  two  fractions  when  each  has  a  numerator 
1  ?— lO-J.  How  do  you  add  mixed  numbers  ? 


COMMON   FKACTIONS. 


137 


When  there  are  mixed  numbers,  add   the   ivhole   numbers 
and  the  fractions  separately,  and  then  add  their  sums. 


Examples. 


1.  Add  J,  j\,  /«,  and  ff 

2.  Add  1,  /^,  if,  tt,  and  M- 

3.  Add  f,  f,  j%  ^S,  and  if. 

4.  Add  J^,  f,  i  and  |. 

5.  Add  1    41    and  f. 

6.  Add  t\,  a,  ¥,  and  §. 

7  Add  -9-    -*-    ^    and  ^ 

8  Add  If,  3i    and  i  of  7. 


9.  AddSff,  7f,  Hand  2]  J. 

10.  Add  2|,  4J,  and  J  of  5jV 

11.  Add  12J,  9|,  f  of  61. 

12.  Add  /_  of  6|and|  of  IJ. 

13.  Add  \  of  9}  and  f  of  4f . 

14.  Addf,i%of  ^\of8,and2i. 

15.  Add  4f,  Vt  of  i  of  l^i- 

16.  Add  3f,  4f,  and  J  of  16. 


17.  Bought  a  cord  of  wood  for  2|-  dollars  ;  a  barrel  of  flour 
for  $9|  ;   and  some  pork  for  15 J  :   what  was  the  entire  cost  ? 

18.  A  person  travelled  in  one  day  35|  miles  ;  the  next,  28^ 
miles  ;  and  the  next  25^y  miles  :  how  many  miles  did  he  travel 
in  the  three  days  ? 

19.  A  grocer  bought  4  firkins  of  butter,  weighing  respective- 
ly 64f,  65f,  51/g,  and  50|^  pounds  :  what  was  their  entu*e 
weight  ? 

20.  I  paid  for  groceries  at  one  time  ^-^  of  a  dollar ;  at  an- 
other, 3 J  dollars  ;  at  another,  7f  dollars  ;  and  at  another,  5^ 
dollars  :  what  was  the  whole  amount  paid  ? 

21.  A  merchant  had  three  pieces  of  Irish  linen  ;  the  first 
piece  contained  22|  yards  ;  the  second  20J  yards  ;  and  the 
third  21 J  yards  :   how  many  yards  in  the  three  pieces  ? 

22.  A  man  sold  5  loads  of  hay  ;  the  first  weighed  18j^^  cwt.  ; 
the  second  lOJi  cwt.  ;  the  third  19§-  cwt. ;  the  fourth  21-J J  cwt.  ; 
and  the  fifth  20 if  cwt.  ;    what  was  the  weight  of  the  whole  ? 

23.  A  farmer  has  three  fields  ;  the  first  contains  17f  acres  ; 


138  AUDITION   OF 

the  second   25-|  acres  ;   and   the  third   463^5  ^cres  :   how  many 
in  the  three  fields  ? 

24.  A  man  sold  112f  bushels  of  wheat  for  250 J  dollars  ; 
9j^2  bushels  of  corn  for  62|-  dollars  ;  225-/j  bushels  of  oats 
for  104  J  dollars  :  how  many  bushels  of  grain  did  he  sell,  and 
how  much  did  he  receive  for  the  whole? 

CASE      II. 

164.    When  the  fractions  have  different  units. 

1.   What  is  the  sum  of  f  lb.  and  f  oz.  ? 

Analysis. — In  operations. 

fib.  there  are  ^oz-  f  1^-  =  t  X  16  oz.  =  ^/  oz. 

(Art.  155.)     Then,  6_4  ^z.  +  f  oz.  =  2^V^  oz.  +  if  oz. 

the    units    of    the  =  2_7_i  ^z.  z=  ISfJ  oz. 
fractions  being  the 

same,   viz.,  1  oz.,   we  reduce  to  a  common  denominator  and  add, 
and  obtain  18^|  oz. 

Second  Method. 
—Three-fourths    of  f  oz.  =  f  X  tV  1^.  =  g^lb. 

an  ounce   is   equal       |  lb.  +  /^-Ib.  ^ffflb.  +  3V0  lb.  =  JJJlb. 
to  ^\  lb.  (Art.  15G.) 

Then,  by  adding,  we  2  ti  i^  =  13-U  oz.  =:  13  oz.  84  dr. 

find  the  sum  to  be 


320  ''^  —  -^"20 


Third     Method.         |  lb.  =  f  X  16  oz.  =  ^3*  oz.  =  12  oz.  12f  dr. 
—Find    the    value         3  q2.  =  s  ^  16  dr.  =  \«  dr.=r  12 

of   each    fractional  Sum         .     .  .     .     13  8*^ 

part    in    terms    of 
integers  of  the  lower  denominations  (Art.  157),  and  then  add. 

Rule. 

I.  Reduce  the  given  fractions  to  the  same  unit,  and  then 
add  as  in  Case  I.     Or, 

II.  Reduce  the  fractions  separately  to  integers  of  lower  de- 
nomijiations,  and  then  add  the  denominate  numbers. 


COMMON    FRACTIONS.  139 


Examples. 

1.  Add  I  of  yard  to  §  of  an  inch. 

2.  Add  together  J  of  a  week,  J  of  a  day,   and  \  of  an 
hour. 

3.  Add  fewt.,  V  ^^-j  1^  oz.,  fcwt.,  and  lib.  together. 

4.  Add  J  of  a  pound  troy  to  J  of  an  ounce. 
6.  Add  ^  of  a  ton  to  j^  of  a  hundredweight. 
6.   Add  f  of  a  chaldron  to  f  of  a  bushel. 

I.  What  is  the  sum  of  f  of  a  tun,  and  f  of  a  hogshead  of 
wine? 

8.  Add  I  of  J  of  a  common  year,  f  of  J  of  a  day,  and  J 
of  §  of  f  of  19^  hours,  together. 

9.  Add  I"  of  an  acre,  f  of  19  square  feet,  and  f  of  a  square 
inch,  together. 

10.  What  is  the  sum  of  -f  of  a  yard,  |  of  a  foot,  aid  ^  of 
an  inch? 

II.  What  is  the  sum  of  |-  of  a  £,  and  f  of  a  shilling? 

12.  Add  together  J  of  a  mile,  f  of  a  yard,  and  f  of  a  foot. 

13.  What  is   the   sum  of  |  of  a   leap  year,    ^    of  a   week, 
and  J  of  a  day  ? 

14.  Add  I  lb.  troy,  A  oz.  and  |  pwt. 

15.  Add  together  y'^   of  a  circle,  3|-  signs,  J   of  a   degree, 
and  f  of  5-}-  minutes. 

16.  What  is  the  sum  of  |-yd.,  f  of  f  qr.  and  3Jna.  ? 

17.  Add  y\  of  a  cord,  f  cubic  feet,  and  |  of  i  of  24f  cubic 
eet. 

18.  What  is   the   sum  of  f  of  i  of  4  cords,  |  of  ^  of  15 
cord  feet,  and  |  of  31 J  cubic  feet? 

19.  Add  I  of  3  ells  English  to  y\  of  a  yard. 


164.  How  do  you  add  fractions  wheii  they  have  difTerent  units? 


140  SUBTRACTION    OF 

20.    Add  together  f  of  3A.  IR.  20  P.,  f  of  an  acre,   and 


f  of  3  K  15  P. 


21.   What  is   the   sum  of  ^^   of  a   ton,  -j^^  of  a  cwt.,  and 


^-  of  an  ounce? 


22.   What  is  the  sum  of  ^  of  f  of  a  mile,  f  of  a  furlong, 
^j  of  a  rod,  and  J  of  a  foot  ? 


SUBTRACTION. 

165.  Subtraction  of  Fractions  is  the  operation  of  finding 
the  difference  between  two  fractional  numbers. 

166.  The  difference  between  two  fractions  is  such  a  num- 
ber as  added  to  the  less  will  give  the  greater. 

CASE   I. 
167.     When  the  unit  of  the  fractions   is  the  same. 

1.  What  is  the  difference  between  J  and  ^  ? 

Analysis. — The   unit  of  both  fractions  is  the  operation. 

same,  being  the  abstract  unit  1.     The  fractional         3        12 
unit  is  also  the  same,  being  ^  in   each;   hence,  4       4       4 

the  difference  of  the  fractions  is  equal  to  the 
difference  of  the  fractional  units,  which  is  f. 

2.  What  is  the  difference  between  |-  lb.  and  f  of  a  pound  ? 

Anai^ysis.— The    unit    in    both  orERATiON. 

fractions    is   lib.      The    fractional  4        2        12        10         2 

unit  of  the  first  is  ]lb.,  and  of  the  5  ~  3  ~  15  ~  15  ~  15 
second  ^Ib.    Reducing  to  the  same 

fractional  unit,  we  have   }f  lb.  and  xI^^m  *^®  difference  of  which 
is  7^5  lb. ;  hence, 

165.  What  is  subtraction  of  fractions  ?— 166.  What  is  the  difference 
between  two  fractions  ?— 167.  How  do  you  subtract  when  the  unit  of 
the  fractions  is  the  same? 


COMMON    FRACTIONS.  141 

Rule. 

I.  If  the  fractional  unit  is  the  same  in  both,  subtract  the 
less  numerator  from  the  greater,  and  i^lace  the  difference  over 
the  common  denominator, 

H.  When  the  fractional  units  are  different,  reduce  to  a 
common  denominator ;  then  subtract  the  less  numerator  from 
the  greater,  and  place  the  difference  over  the  common  denom- 
inator. 

Examples. 


1.  From  f  take  \, 

2.  From  \^  take  -jj. 

3.  From  Jf  take  ^|. 

4.  From  joj  take  ij|. 

5.  From  f  take  f. 

6.  From  \^  take  |f. 

7.  From  j|  take  |J. 

8.  From  37 }i  take  J  of  5J-. 


9.  From  f  take  |. 

10.  From  J  take  jV 

11.  From  25  take  jj. 

12.  From  /^  of  3  take  J  of  |. 

13.  From  i  of  f  of  7  take  f. 

14.  From  3f  take  f  of  f 

15.  From  f  of  15  take  |  of  3. 

16.  From  7  J  of  2  take  J  off. 


17.  To  what  fraction  must  I  add  |  that  the  sum  may  be  f  ? 

18.  What  number  added  to  1|^,  will  make  5  ? 

19.  What  number  is  that  to  which  if  7 J  be  added  the  sum 


will  be  17f  ? 


20.  From  the  sum  of  3|-  and  10|-  take  the  difference  of  25} 
and  17iJ. 

21.  What  number  is  that  from  which  if  you  subtract  J  of 
I  of  a  unit,  and  to  the  remainder  add  f  of  J  of  a  unit,  the 
sum  will  be  9  ? 

22.  If  I  buy  f  of  I  of  a  vessel,  and  sell  J  of  .|  of  my 
share,  how  much  of  the  whole  vessel  will  I  have  left? 

23.  A  man  bought  a  horse  for  ^  of  |  of  ^^  of  $500,  and 
sold  him  again  for  f  of  J  of  f  of  $1680  :  what  did  he  gain 
by  the  bargain  ? 


143  SUBTRACTION   OF 

24.  Bought  wheat  at  1|-  dollars  a  bushel,  and  sold  it  for 
21  dollars  a  bushel  :  what  did  I  gain  on  a  bushel  ? 

25.  From  a  barrel  of  cider  containing  31^  gallons,  12f  gal- 
Ions  were  drawn  :  how  much  was  there  left  ? 

26.  Bought  lOJ  cords  of  wood  at  one   time,  and  24|  cords 
t  another;   after  using  16 J  cords,  how  much  remained? 

2Y.  A  merchant  bought  two  firkins  of  butter,  one  contain 
ing  54j%  pounds,  and  the  other  56}J  pounds  ;  he  sold  43}! 
pounds  at  one  time,  and  34  J  pounds  at  another :  how  much 
had  he  left? 

28.  A  man  having  $50  J,  expended  $15y^^  for  dry-goods,  and 
12|-  for  groceries  :  how  much  had  he  left? 

29.  A  boy  having  f  of  a  dollar,  gave  J  of  a  dollar  for  aa 
inkstand,  and  J  of  it  for  a  slate  :  how  much  had  he  left  ? 

30.  Bought  two  pieces  of  cloth,  one  containing  27f  yards, 
the  other  32 1-  yards,  from  which  I  sold  40^  yards :  how  much 
had  I  left? 

168.     When  each  fraction  has  the  numerator  1. 

1.   What  is  the  difference  between  J  and  l? 

Analysis. — Reducing  both   frac-  operation. 

tions    to    a    common    denominator         ^  —  l-  =  -A.  —  -^  =  -^. 
and  subtracting,  we  find  the  difier- 
ence  to  be  -^^ ;  that  is, 

The  difference  between  two  fractions,  each  of  whose  numer- 
ators is  1,  is  equal  to  the  difference  of  the  denominators 
divided  by  their  product. 


2.  From  J  take  yV- 

3.  From  Jj  take  jL. 


4.  From  jL  take  ^, 

5.  From  Jy  take  ^. 


168.  What  is  the  difference  when  the  numerator  of  each  fraction 
is  1? 


COMMON    FRACTIONS.  143 


169.    WTien  there  are  mixed  numbers. 

1.  What  is  the  difference  between  16J  and  3J  ? 
Analysis. — Since  we  cannot    take  y%  from  operation. 

•^\,  we  borrow   1  =  y|  from  the  whole  num-         16J  =  IGjV 
ber  of  the  minuend,  which,  added  to  fV,  gives  3J  =    3j\. 

|-f :  then  j^j  from  -}-|  leaves  j§.     We  must  now  T24^. 

carry  1  to  the  next  figure  of  the  subtrahend, 
and  say  4  from  16  leaves  12.     Hence,  to  subtract  one  mixed  num- 
ber from  another, 

Subtract  the  fractional  part  from  the  fractional  part,  and 
the  integral  part  from  the  integral  part. 

2.  What  is  the  difference  between  144-  and  12^%? 

3.  What  is  the  difference  between  115f  and  39 J? 

4.  What  is  the  difference  between  ^^y&  ^^^  ^'h'^- 

5.  What  is  the  difference  between  48 jV  and  41J-|? 

6.  What  is  the  difference  between  287^  and  104/^^^  ? 

CASE    II. 
170.    When  fractions  have  different  imits. 

1.   What  is  the  difference  between  J  of  £  and  J  of  a  shil- 
Hng  ?  ,. 

Analysis. — Reducing  to  the 
common  unit  Is.,  we  find  the 

difference     to     be     V^-  =  ^s-        V^-  —  ¥-  =  %°s.  —  §s.  =  ^S- 
8d. 

Second    Method. — Reducing 
to   the    common    unit   £1,   we 

find  the  difference  to  be  £|^  =  _  ^2_9  _  gg^  gj^ 

9s.  8d. 

TniRD  Method. — Reduce  the  £1  =  lOs. 

fractions  to  integral  units,   and  Js.  —  4d. 

then  subtract  as  in  denominate  "^sTScL 

numbers. 


OPEEiHON. 

^ 

=  i  X  20s.  : 

Vs- 

-  Js.  =  t»s.  . 

-|s.= 

=  9|s.  =  9s. 

8d. 

4s.= 

Jx£,V  = 

£A- 

£h- 

JE^  =  £f  ? 

-^- 

144  SUBTRACTION    OF 


Rule. 

I.  Reduce  the  fractions  to  the  same  unit,  and  then  subtract 
as  in  Case  I,:     Or, 

II.  Flyid  the  value  of  each  fraction  in   units  of  lower  de- 
nominations, and  then  subtract  as  in  denominate  numbers. 


Examples. 

1.  From  f  of  a  pound  troj,  take  f  of  an  ounce. 

2.  From  f  of  a  ton,  take  f  of  J  of  a  pound. 

3.  From  -|  of  f  of  a  hogshead  of  wine,  take  f  of  J  of  a 
quart. 

4.  From  f  of  a  league,  take  f  of  a  mile. 

5.  What  is  the  difference  between  Ifs.  and  f  of  tjd? 

6.  What  is   the   difference   between  §i  of  a  degree  and  f 
of  j-  of  a  degree. 

7.  From  i|  of  a  square  mile,  take  36J  acres. 

8.  From  f  of  a  ton,  take  f  of  12  cwt. 

9.  From  If  lb.  troy,  take  \  of  an  ounce. 

10.  From  2f  cords,  take  f  of  a  cord  foot. 

11.  From  1  of  a  yard,  take  f  of  an  inch. 

12.  From  i  of  J  of  a  pound,  take  f  of  J  of  a  dram,  apoth- 
ecaries' weight. 

13.  From  a  piece  of  ground  containing  2^^%  acres  take  lA. 
IP.  and  9  square  yards. 

14.  A  pound  avou'dupois  is  equal  to  14  oz.  llpwt.  IGgr. 
troy  :  what  is  the  difference,  in  troy  weight,  between  the  ouncG 
avoirdupois  and  the  ounce  troy? 


1G9.    How   do  you   subtract   when    there    are    mixed    numbers? 
170.  What  is  the  rule  wlien  tlie  fractions  have  different  units. 


COMMON    FK ACTIONS.  145 


MULTIPLICATION. 

171.  Multiplication  of  Fractions  is  the  operation  of  taking 
one  number  as  many  times  as  there  are  units  in  another,  when 
one  or  both  are  fractional. 

I.  If  1  pound  of  tea  cost  f  of  a  dollar,  what  will  ^*of  a 
pound  cost. 

Analysis. — The  cost  will  he  equal  operation. 

to  the  price  of  1  lb.  taken  as  many         $|.  X  y  =  g  ^  |  =  $|-|* 
times  as  there  are  units  in  the  multi- 
plier (Art  84). 

One-seventh  of  a  pound  of  tea  will  cost  one-seventh  as  much  aa 
lib.  Since  lib.  cost  $f,  |  of  lib.  will  cost  }  of  $f  =  $5^  (^rt. 
142).  But  3  sevenths  of  lib.  will  cost  three  times  as  much  as 
I ;  that  is,  $/g^  x  3  =  ^|  (Art.  139).  Hence,  to  multiply  one  frac- 
tion by  another: 

Rule. 

Cancel  all  factors  common  to  the  numerator  and  denom- 
inator;  then  multiply  the  numerators  together  for  a  new 
numerator,  and  the  denominators  together  for  a  new  denom- 
inator, 

172.    Principles  of  the  operation. 

1.  When  the  multiplier  is  less  than  1,  we  do  not  take  the  wkole 
of  the  multiplicand,  but  only  such  a  part  of  it  as  the  multiplier  is 
of  1. 

2.  When  tbe  multiplier  is  a  proper  fraction,  multiplication  does  not 
increase  tbe  multiplicand,  as  in  the  multiplication  of  whole  numbers. 
The  product  is  the  same  part  of  the  multiplicand  as  the  multiplier 
is  of  1. 

8.  ^Vhen  either  of  the  factors  is  a  whole  number,  write  1  under  it 
for  a  denominator. 

4.  \Vhen  either  of  the  factors  is  a  mixed  number,  reduce  it  to  an 
improper  fraction. 

171.  Wliat  is  multiplication  of  fractions?  What  is  the  rule?— 173 
What  is  the  first  principle  of  the  operation?    What  is  the  second? 

7 


146 


MULTIPLICATION   OF 


1.  Multiply  f  by  8. 

2.  Multiply  j\  by  12. 

3.  Multiply  Jf  by  9. 

4.  Multiply  If  by  15. 

9.  Multiply  6T  by  9^2- 

10.  Multiply  842  by  TJ. 

11.  Multiply  360  by  12f. 

15.  Multiply  f  by  8. 

16.  Multiply  15  by  f. 
11.  Multiply  1J  by  8. 

18.  Multiply  91  by  18|. 

19.  Multiply  3|-  by  4if. 


Examples. 

5.  Multiply  I  of  A  by  35. 

6.  Multiply  If  of  2 i  by  16. 
1.  Multiply  21  of  f  by  TO. 
8.  Multiply  4|  of  8  hj  36. 

12.  Multiply  4G0  by  llf. 

13.  Multiply  620  by  lOf. 

14.  Multiply  1340  by  8|. 

24.  Multiply  ^  of  f  by  f  of  j%. 

25.  Multiply  f  by  16. 

26.  Multiply  28  by  y\. 
21.  Multiply  8/o  by  15. 
28.  Multiply  ^t  of  f  by  if. 


V¥  by  9. 


20.  Multiply 

21.  Multiply  I  by  |. 

22.  Multiply  J  of  f  by  |. 

23.  Multiply  ^^  by  2^^  of 


33.   Multiply  j\,  If,  and  -,-, 


2  7* 
46 


29.  Multiply  5J  by  |  of  3. 

30.  Multiply  8421  by  IJ. 

31.  Multiply  f  by  f. 


32.   Multiply  j%  by  1j\. 

together. 

34.  Multiply  ^f  ^^,  ^,  aud  ff  together. 

35.  What  is  the  product  of  }f  by  f  of  17. 

36.  What  is  the  product  of  6  by  |  of  5. 

3t.  What  is  the  product  of  J  of  J  of  3  by  15|  ? 

38.  What  is  the  product  of  f  of  f  by  f  of  3f  ? 

39.  What  is  the  product  of  5,  f ,  |  of  f ,  and  41  ? 

40.  What  will  1  yards  of  cloth  cost,  at  $f  a  yard  ? 

41.  What  will  12|  bushels  of  apples  cost,  at  If-  a  bushel  ? 

42.  If  one   bushel   of  wheat   costs  |1|,  what  will  |    of  a 
bushel  cost  ? 


COMMON    FRACTIONS.  14? 

43.  If  one  horse  cats  J-  of  a  ton  of  hay  in  one  month,  how 
much  will  18  horses  eat  iu  the  same  time? 

44.  If  a  man  earns  ^f|  in  one  day,  how  much  can  he  earn 
in  24  days  ? 

45.  What   will   3J  yards   of  cloth  cost,  at  J  of  a  dollar  a 
yard  ? 

46.  At  $16  a  ton,  what  will  \^  of  a  ton  of  hay  cost? 

47.  If  one   pound   of  tea   costs   $1J,  what  will   6J   pounds 
cost? 

48.  What  will  3f  boxes  of  raisins  cost,  at  $2J  a  box  ? 

49.  At  75  cents  a  bushel,  what  will  fi  of  a  bushel  of  corn 
cost  ? 

50.  If  a  lot   of  land  is  worth  $75j®j,  what  will  j\  of  it  be 
worth  ? 

51.  What  will  17 J  yards  of  cambric  cost,  at  2 J  shillings  a 
yard? 

52.  Bought   15f   barrels   of  sugar,  at   $20  J   a  barrel :  what 
did  the  whole  cost  ? 

63.   If  one  bushel  of  corn  is  worth  f  of  a  dollar,  what  is  | 
of  a  bushel  worth  ? 

54.  If  I  own  ^  of  a  farm,  and   sell  ^  of  my  share,  what 
part  of  the  whole  farm  do  I  sell  ? 

55.  I  bought  a  book   for  ^q   of  a  dollar,  and  a  knife  for 
,*^  the  cost  of  the  book  :   how  much  did  I  pay  for  the  knife  ? 

56.  At  }  of  fj  of  a  dollar  a  pound,  what  will   J^  of  |f  of 
a  pound  of  tea  cost? 

57.  If  hay  is  worth  $9|-  a  ton,  what  is  J  of  3 J  tons  worth? 

58.  If  a  man  can  dig  a  cellar  in  22 J  days,  how  many  day 
ould  it  take  hun  to  dig  f  of  it  ? 

59.  If  a  railroad  train  runs  1  mile  in    ^  of  an   hour,  how 
tuiig  will  it  be  in  running  106 J  miles  ? 


148  DIVISION    OF 

60.  A  owned  I  of  a  farm  and  sold  f  of  his  share  to  B, 
who  sold  f  of  what  he  bought  to  C,  who  sold  f  of  what  he 
bought  to  D  :  what  part  of  the  whole  did  D  have  ? 

61.  A  owned  f  of  200  acres  of  land,  and  sold  f  of  his  share 
to  B,  who  sold  J  of  what  he  bought  to  C  :  how  many  acres 
had  each  ? 

DIVISION. 

173.  Division  of  Fractions  is  the  operation  of  finding  how 
many  times  one  number  is  contained  in  another,  when  one  or 
both  are  fractional. 

1.   What  is  the  quotient  of  J  divided  by  y  ? 

Analysis. — How  many  times  is  ^/  operation. 

contained  in  |  ?     If  I  be  divided  by         1  ^  y  =  J  ^  ^-, 
14,   the  quotient  will  be  .g^  =  J_.  =  _5_  j^^g^ 

Since  the  true  divisor  is  but  }  of  14, 

the  divisor  used  is  5  times  too  large;  hence,  the  partial  quotient 
y'g,  is  5  times  too  small.  Multiplying  this  by  5,  we  have  the  true 
quotient,  =  ■^\.  This  result  is  produced  by  inverting  the  terms  of 
the  divisor  and  multiplying. 

Rule. — Invert  the  terms  of  the  divisor,  cancel,  and  proceed 
as  in  multiplication. 

174.    Directions  for  the  operation. 

1.  If  either  the  dividend  or  divisor  is  a  whole  number,  make  it 
fractional,  by  writing  1  under  it  for  a  denominator. 

2.  Cancel  all  common  factors. 

3.  If  the  dividend  and  divisor  have  a  common  denominator,  they 
will  cancel,  and  the  quotient  of  the  numerators  will  be  the  answer. 


173.  What  is  division  of  fractions?  What  is  the  rule?— 174.  What 
is  the  first  direction  for  performing  the  operation?  What  the  second? 
What  the  third?    What  the  fourth?    W^hat  the  fifth? 


COMMON     FRACTIONS. 


U9 


4.  When  either  term  of  the  fraction  is  a  mixed  number,  reduce  to 
the  form  of  a  simple  fraction. 

5.  If  the  numerator  of  the  dividend  is  divisible  by  the  numerator 
of  the  divisor,  and  the  denominator  by  the  denominator,  divide  with- 
out inverting. 


1. 

Exai] 
Divide  f}  by  1. 

aple 
26. 

3. 

Divide  jf  by  4. 

2. 

Divide  ^  by  6. 

27. 

Divide  Jf  by  5. 

3. 

Divide  if  by  9. 

28. 

Divide  fa  by  8. 

4. 

Divide  Jfg  by  40. 

29. 

Divide  i^}  by  48 

5. 

Divide  ff  by  13. 

30. 

Divide  j\^^  by  21 

6. 

Divide  5  by  ^^. 

31. 

Divide  36  by  j%. 

1. 

Divide  27  by  -J. 

32. 

Divide  420  by  |. 

8. 

Divide  i  by  i. 

33. 

Divide  /^   by  f. 

9. 

Divide  j%  by  f . 

34. 

Divide  iJ  by  j\. 

10. 

Divide  U  ^y  tV 

35. 

Divide  f  of  f J  by   |f. 

11. 

Divide  f  of  f  by  f  of  f 

36. 

Divide  I  by  if. 

12. 

Divide  l  of  |  by  |  of  f . 

37. 

Divide  f  of  f  by  f  of  f . 

13. 

Divide  1  of  f  by  f  of  |. 

38. 

Divide  i  of!  off  by!  of  |. 

14. 

Divide  56  by  ji 

39. 

Divide  650  by  i^f 

15. 

Divide  1000  by  -^j. 

40. 

Divide  1273  by  i|. 

16. 

Divide  725  by  Jf 

41. 

Divide  4324  by  Jff. 

n. 

Divide  4|-  by  5. 

42. 

Divide  6f  by  8. 

18. 

Divide  9xV  by  12. 

43. 

Divide  12f  by  42. 

19. 

Divide  J  of  16J  by  41. 

44. 

Divide  3i  by  9|. 

20. 

Divide  9J  by  J  of  7. 

45. 

Divide  100  by  4f. 

21. 

Divide  f  of  50  by  4^. 

46. 

Divide  443V  by  |i§. 

22. 

Divide  300/^  by  6i 

47. 

Divide  111  J  by  33^. 

23. 

Divide  4  of  3f  by  ij  of  7^. 

48. 

Divide  191}  by  159}. 

24. 

Divide  9|  by  SJ. 

49. 

Divide  5f  by  f  of  1^. 

25. 

Divide  1  of  -j^  by  6^. 

50 

Divide  5205}  by  f  of  90. 

160  DIVISION  OF 

51.  At  i   of  a  dollar  a  pound,   how  much  butter  can  be 
bought  for  I J  of  a  dollar  ? 

52.  At  f  of  a  dollar  a  yard,  how  much  cloth  can  be  bought 
for  I"  of  a  dollar  ? 

53.  If  a   bushel  of  potatoes   cost   |  of  a  dollar,  how  many 
bishels  can  be  bought  for  ^^  of  a  dollar? 

64.  If  I  of  a  ton  of  hay  will  feed   1  horse  one  week,  how 
many  horses  will  -j^  of  a  ton  feed,  the  same  time  ? 

55.  If  f  of  a  bushel  of  apples  cost  f  of  a  dollar,  what  will 
1   bushel  cost  ? 

56.  What  will  a  barrel  of  flour  cost,  if  ^g-  of  a  barrel  cost 
I  of  a  dollar  ? 

57.  If  f  of  a  bushel  of  apples  cost  f  of  a  dollar,  what  will 
1  bushel  cost  ? 

58.  How  much   molasses  at   f   of  a  dollar  a  gallon,  can  be 
bought  for  1^  dollars  ? 

59.  A  man  sold   fj  of  a  mill,  which  was  J  of  his   share  : 
what  part  of  the  mill  did  he  own  ? 

60.  What  number  multiplied  by  f,  will  give  a  product   of 
15f? 

61.  What  number  multiplied  by  5 J,  will  give  a  product  of 
146? 

62.  The  dividend  is   620 J,  and  the  quotient   36/o  :   what  is 
the  divisor  ? 

63.  What  number  is  that,  which  if  multiplied  by  f  of  f   of 
I5|-,  will  produce  f  ? 

64.  If  71b.  of  sugar  cost  |-f  of  a  dollar,  what  will  1  pound 
cost? 

65.  If  10  J  lb.  of  nails  cost  f  of  a  dollar,  what  is  the  price 
per  pound  ? 


COMMON   FllAGTIONS.  151 

C6.    If  f  of  a  yard  of  cloth  cost  $3,  what  will  1  yard  cost  ? 

67.  A  family  consumes  165f  pounds  of  butter  in  8J  weeks : 
how  much  do  they  consume  in  1  week  ? 

68.  At   $9|   a  barrel,   how  much   flour  can  be   bought  for 

nssf  ? 

69.  If  a  man  divides    $3|-  equally   among   8   beggars,   how 
much  docs  he  give  them  apiece  ? 

70.  If    8  pounds  of   tea  cost  $7f,   what   is   the  price  per 
pound  ? 

71.  If  J  of  a  ton  of  hay  sells  for  $10 J,  what  is  the  price 
of  1  ton  ? 

72.  If  J   of  an  acre   of  ground  produces   84/^  bushels   of 
potatoes,  how  many  bushels  will  1  acre  produce? 

73.  What  quantity  of  cloth   may  be  purchased  for  $5j'g,  at 
the  rate  of  $6f  a  yard  ? 

74.  How  long  would  a  person  be  m  traveling   125|-  miles, 
if  he  traveled   31y^  miles  per  day? 

75.  How   many   bottles,   each    holding    If    gallons,   can   be 
filled  from  a  barrel  of  wine,  containing  31 J  gallons? 

76.  How  long  will   it  take  11  men  to  do  a  piece   of  work, 
that  1  man  can  do  in  15|  days? 

77.  If  f  of  a  barrel  of  flour  costs  6  dollars,  what  is   the 
price  per  barrel  ? 

78.  Eighty-one  is  f  of  how  many  times  8  ? 

79.  Five-eighths  of  48  is  f  of  how  many  times  9  ? 

80.  How  many  times  can  a  vessel,  containing  |  of  a  gallon, 
be  filled  from  J  of  a  barrel  of  31^  gallons? 

81.  If  5^ lb.  of  tea  cost  $4f,  what  is  the  price  of  1  pound? 

82.  If  f  of  f  of  a  ship  is  worth  $2540,  what  is  the  whole 
vessel  worth? 

83.  If  f  of  f  of  a  barrel  of  flour  willlast  a  family  1  week, 
how  long  will  9j\  barrels  last  them? 


152 


COMPLEX  FRACTIONS. 


COMPLEX    FRACTIONS. 
175.   A  Complex  Fraction  is  only  another  form  of  expression 

7. 

for  the  division  of  fractions  :  thus,  5.,  is  the  same  as  ^  divided 
^y  f  ;  and  may  be  written, 


5.  —  4  2 

6   —   45' 


176.    To  reduce  a  complex  fraction  to  a  simple  fraction. 
6i 


1.   Reduce  _3  to  a  simple  fraction. 


Analysis.  —  Eeducing      th  e 
divisor  and  dividend   each   to 


OPERATION. 

6}  =  2_o^  and  11  =  f. 
a  simple  fraction,  we  have  ^  2_o  _u  8  _  2_o  v  1  —  3_5  — 
and  f .  Then  %'  divided  byf  ^  •  i  —  3  ^  s  —  e  — 
is  equal  to  %°  x  ^  =  V  =  H- 

Rule. — Beduce  both  terms  of  the  fraction  to   simple  frajo- 
lions:  then  divide  as  in  division  of  fractions. 

Examples. 
Reduce  the  following  to  simple  fractions : 

t.   Reduce  HA. 

87 


1. 

Reduce   «. 

i 

2. 

Reduce  X. 
if 

3. 

Reduce  It. 

4. 

Reduce  ^^^. 

6. 

Reduce  X. 

6. 

Reduce  ?i. 
12 

8.   Reduce 


20 

4  ' 

T 


9. 

Reduce 

f    of    1^\ 

xVofnf 

0. 
1. 

Reduce 
Reduce 

26A 
1  of  17' 

55A 

12.  Reduce  f  of /^  o^^- 

8  10  jg 


175.  Wliat    is    a    complex   fraction? — 176.  How   do    you   reduce   a 
complex  to  a  yhiiple  fraction? 


APPLICATIONS.  153 

Applications  in  Fractions. 

1.  What  will   5^-  cords  of  wood  cost,  at  i  of  f  of  |  of 
150  a  cord? 

2.  A  farmer  sold  |  of  a  ton  of  hay  for  $Gf  :  what  would 
be  the  price  of  a  ton  at  the  same  rate  ? 

3.  A  person  walks  7tf  miles  in  10 J  hours  :   at  what  rate 
is  that  per  hour? 

4.  From  the  product  of  f  and  11  J,  take  ^,  and  multiply 
the  remainder  by  20 J. 

5.  How  much  greater  is  f  of  the  sum   of  J,  J,  1,  and  J, 
than  the  sum  of  J,  J,  and  J  ? 

6.  If  I  of  a   ton   of  hay  is  worth  $Ti   what  is   2|   tona 
worth  ? 

T.   If  f  of  a  dollar  will  pay  for  |  of  a  yard  of  cloth,  how 
many  yards  can  be  bought  for  $llf  ? 

8.  What  is  the  value  of  3 J  cords  of  wood,  at  $4§  a  cord  ? 

9.  At  J  of  a  dollar  a  peck,  how  many  bushels   of  apples 
can  be  bought  for  $6}? 

10.  AVhat  is  the  difference  between  J  of  a  league  and  ^^^ 
of  a  mile  ? 

11.  What  is  the  sum  of  4^^  miles,  |-  of  a  furlong,  and  | 
of  1 J  yards  ? 

12.  At  $1J  per  day,  how  many  days^  labor  can  be  obtained 
for  $3Gf  ? 

13.  Bought  5 J  yards  of  cloth  at  $4 J  a  yard,  and  paid  for 
it  in  wheat  at  ^l-f  a  bushel :  how  many  bushels  were  required  ? 

14.  What  number  must  be  taken  from  2*r|,  and  the  re- 
mainder multiplied  by  14f ,  that  the  product  shall  be  100  ? 

15.  Three  persons.  A,  B,  and  C,  purchase  a  piece  of  prop- 
erty for  $6300  ;  A  pays  f  of  it,  B  J,  and  C  the  remainder  : 
what  is  the  value  of  each  one's  share  ? 


154  COMPLEX   P^KACTIONS. 

16.  What  number  diminished  by  the  difference  between  f 
and  f  of  itself,  leaves   a  remainder   equal  to   34  ? 

IT.  What  is  the  sum  of  f  of  ^£15,  £^,  J  of  f  of  f  of 
£1,  and  f  of  f  of  a  shilling? 

18.  If  ^  of  John's  marbles  is  equal  to  J  of  James\  and 
together  they  have  66,  how  many  has  each? 

19.  A  person  owning  f  of  2000  acres  of  land,  sold  |  o. 
his  share  :  how  many  acres  did  he  retain  ? 

20.  A  boy  having  240  marbles,  divided  them  in  the  follow- 
ing manner  :  he  gave  to  A,  J,  to  B,  j^,  to  C,  J,  and  to  D, 
J,  keeping  the  remainder  himself :  what  number  of  marbles 
had  each? 

21.  A  man  having  engaged  in  trade  with  |3t40,  found,  at 
the  end  of  3  years,  that  he  had  gained  $156  J  more  than  J  of 
his  capital :  what  was  his  average  annual  gain  ? 

22.  Two  boys  having  bought  a  sled,  one  paying  |  of  a  dollar, 
and  the  other  J  of  a  dollar,  sold  it  for  ^^  of  a  dollar  more  th'in 
they  gave  for  it :  what  did  they  sell  it  for,  and  what  was  each 
one's  share  of  the  gain  ? 

23.  A  farmer  having  126f  bushels  of  wheat,  sold  f  of  it  at 
|2i  a  bushel,  and  the  remainder  at  Ufa  bushel  :  how  much 
did  he  receive  for  his  wheat  ? 

24.  A  man  having  $19^,  expended  it  for  wheat  and  corn,  of 
Q-dch  nn  equal  quantity  ;  for  the  wlieat  he  paid  81|  a  l)u.^]iel, 
nn  !   foi"  tlic  corn  ||  a  bushel  :  how  mucli  of  each  did  he  buy? 

25.  Two    jiersoiis    engage    in    trade:    A   furnislied   /.-j   -''     '■ 
.'.'ipltal,    and    13,    -j\  :     if   B    had    furnished    *492|    ui  • 
shiires  would  have  been  equal  :  how  much  did  each  furn;  -^i  . 

26.  A  man  being  asked  how  many  sheep  he  had,  said,  he  had 
them  in  three  fields  :  m  the  first  he  had  63,  which  was  |-  of 
what  he  had  in  the  second ;  and  |  of  what  he  had  in  the  -ecoiid 
was  4  times  what  he  had  in  the  third  :  how  many  had  he  in  all  ? 


DUODKCIMALS.  155 


DUODECIMALS. 

177.  DuoDEcmALS  are  a  system  of  numbers,  which  arise  from 
dividing  a  unit  according  to  the  scale  of  12.  The  units 
divided  arc,  the  foot  in  length,  the  square  foot,  and  the  cubic 
foot. 

If  the  unit  1  foot  be  divided  into  12  equal  parts,  each 
part  is  called  an  inch  or  prime,  and  marked '.  If  a  prime  be 
divided  into  12  equal  parts,  each  part  is  called  a  second,  and 
marked".  If  a  second  be  divided,  in  like  manner,  into  12 
equal  parts,  each  part  is  called  a  third,  and  marked '"  ;  and 
so  on  for  divisions  still  smaller  :  hence, 

y^^  of  a  foot  —-  1  inch,  or  prime, 1'. 

tV  0^  T2  ^^  ^  ^^^^  =  T4T  ^^  ^  ^o^*>  ^^  1  second,     .     1''. 
tV  ^^  tV  ^^  tV  ^^  ^  ^0^^  =  TT2T  0^  ^  ^oo*»  ^^  ^  third,  .     1'" 

If  the  square  foot,  and  the  cubic  foot,  be  divided  according 
to  the  same  scale,  the  primes,  seconds,  thirds,  &c.,  will  have 
the  same  relation  to  the  unit  and  to  each  other,  as  in  the 
foot  of  length. 

Table. 

12'" make   1"   second. 

12" "       1'    inch  or  prime. 

12' "       1     foot. 

Hence  :  Duodecimals  are  denominate  fractions,  in  which  the 
primary  unit  is  1  foot,  and  the  scale  uniform,  the  units  of  the 
scale,  at  every  point,  being  12. 

Notes. — 1.  The  marks',  ",  '",  &c.,  which  denote  the  fractional 
units,  are  called  indices. 

2.  Duodecimals  are  chiolly  used  in  measuring  LengtJis,  Surfaces, 
Volumes,  or  Solids. 

177.  Wliat  are  duodecimals?  What  are  tlie  units  divided?  If  the 
unit  1  foot  be  divided  into  12   oi[ual    partes  what  is  each  part  called? 


156  DUODECIMALS. 

ADDITION    AND    SUBTRACTION. 

178.  The  operations  of  Keduction,  Addition,  Subtraction, 
Multiplication,  and  Division  of  Duodecimals,  correspond  so 
nearly  with  those  of  denominate  numbers,  that  additional  rules 
are  deemed  unnecessary. 

Examples. 


1.  In  86'  how  many  feet? 

2.  In  t50"  how  many  ft.? 

3.  In  3t000"'howmanyft.? 


4.  In  6V  how  many  feet? 

5.  In  4Y0'"  how  many  ft.? 

6.  In  375"  how  many  ft.? 
1.  What  is  the  sum  of  8  ft.  9'  V  and  6  ft.  Y  3"  4'"? 

8.  Find  the  difference   between  32  ft.  6'  6"  and  29  ft.  1'" 

9.  Add  together  9  ft.  6'  4"  3'",  12  ft.  2'  9"  10'",  26  ft.  0' 
5",  and  40  ft.  1'  0"  3"\ 

10.  What  is  the  sum  of  125  ft.  0'  6",  45  ft.  11'  0"  2"', 
and  12  ft.  6'? 

11.  What  is  the  sum  of  84  ft.  r,  96  ft.  0'  11",  42  ft.  6'  9" 
10''',  and  5'  1"  11'"? 

12.  From  12Ut.  3'  6"  4'"  11"",  take  40  ft.  0'  10"  V"  5"". 

13.  What  is  the  difference  between  425  ft.  9'  10"  and  107  ft. 
10'  9"  8'"  ? 

14.  What  is  the  sum  and  difference  of  325  ft.  7'  6"  2'"  and 
217  ft.  10'  9"? 

MULTIPLICATION. 

179.  Multiplication  of  Duodecimals  is  the  operation  of  find- 
ing the  superficial  contents  and  the  contents  of  volume,  when 
the  linear  dimensions  are  known. 

If  1  inch  be  divided  into  12  equal  parts,  what  is  each  part  called 
If  the  seccaid  be  divided  in  like  manner,  what  is  eacli  part  called 
What  are  indices?    For  what  are  duodecimals  used? — 178.   How  are 
the  fundamental  operations  performed? — 179.  Wliat  is  multiplication 
of  duodecimals ?    How  are  the  areas   of  figures  found?    How  are  the 
•contents  of  volume  found  ? 


MULTIPLICATION. 


157 


The  superficial  contents,  or  area  of  figures,  are  found  by 
multiplying  the  length  and  breadth  together. 

The  contents  of  volume  or  cubical  contents,  arc  found  by  mul- 
tiplying together  the  length,  breadth,  and  height. 


180.    Principles  of  the  Multiplication. 

1.  Feet  multiplied  by  feet,  give  square  feet. 

2.  Feet  x  Primes  =  1  ft.  x  j\  ft.  =  j\  sq.  ft.,  or  primes. 

3.  Primes  x  Primes  =  yV  ft.  x  i  ^t*  =  tIi  sq.  ft.,  or  seconds. 

4.  Primes  x  Seconds  =  yV  ft.  x  yii  ^**  =  tt*2^  s^*  ^*->  ^^  thirds. 

5.  Seconds  x  Seconds  =  y|j  ft.  x  j\^  ft.  =  g^TSff  ^q.  ft.,  or  fourths. 

From  the  foregomg,  we  have  the  following  principles  : 

17ie  index  of  any  product  is  equal  to  the  sum  of  the  in- 
dices of  the  factors. 

Note. — The  denominator  of  primes  is  13,  of  seconds,  144,  of  thirds 
1728,  of  fourths,  2073G,  &c. 

181.    To  find  the  square  measure,  or  area  of  a  surface. 

1.    Find   the   square   measure   of  a  floor  that  is  9  feet  long 
and  G  feet  wide. 

Note. — A  Square  is  a  figure  bounded  by  four  equal  sides  at  right 
angles  to  each  other. 


Analysis. — Draw  an  horizontal  line 
and  lay  off  9  equal  parts,  each  de- 
noting a  foot.  Then  draw  a  second 
horizontal  line  perpendicular  to  it, 
and  lay  off  6  equal  parts,  each  denot- 
ing a  foot.  Through  the  points  of 
division   of  the  first  lino  draw  paral-  9 

els    to   tlio  second,    and    through    the    points    of  division    of  the 


IbO.  \Vhi\t  are  the  five  principles  of  multiplication?  What  is  the 
rulo  for  the  indices?  What  is  the  rule  for  the  multiplication  of  duo- 
decimals?—181.  What  is  the  rule  for  finding  the  square  measure  of 
a  surface? 


158 


DUODECIMALS. 


second  line  draw  parallels  to  the  first:  there  will  thus  be  formed 
number  of  small  squares. 

The  number  of  squares  in  the  first  row  will  be  equal  to  9,  the 
number  of  linear  units  in  the  first  line;  and  the  number  of  rows 
v>i]l  be  equal  to  six,  the  number  of  units  in  the  second  line :  there- 
fore, the  whole  number  of  squares  will  be  equal  to  9  x  6  =  54. 
Hence,  to  find  the  area,  or  measure, 

IfliiUiply  the  length  by  the  breadth,  and  the  product  will  6e« 
the  number  of  square's. 

Note. — The  square  which  is  the  unit  of  surface,  is  the  square  de- 
scribed on  the  unit  of  length.  If  the  unit  of  length  is  a  foot,  the 
unit  of  surface  is  1  square  foot — if  1  yard,  the  unit  of  surface  is  1 
square  yard,  &c. 


182.  To  find  the  Cubic  Measure  of  a  Volume  or  Solid. 

1.  What  is  the  cubic  measure  of  a  block  of  marble  that  is 
9  ft.  long,  6  ft.  wide,  and  4  ft.  thick  ? 

Note. — A  Cube  is  a  figure  bounded  by  six  equal  squares  at  right 
angles  to  each  other,  called  faces;  and  the  sides  of  the  squares  are 
the  edr/cs  of  the  cube. 


Analysis.  —  The  face  on  which 
the  block  stands,  is  called  its  dase, 
tlie  area  of  which  is  equal  to  9  x  6 
=  54  sq.  ft. 


If  now  you  take  54  equal  cubes,  of  1  foot  each,  they  can 
be  placed  side  by  side  on  the  base,  and  will  form  a  block  of 
marble  9  ft.  long,  6  ft.  wide,  and  1  foot  thick.  If  you  place 
a  second  tier,  the  block  will  be  2  feet  thick ;  a  third  tier  will 
make  it  8  feet  thick,  and  so  on,  for  any  number  of  tiers: 
hence,  the  contents  of  the  block,  that  is  four  feet  thick,  are 
9  X  G  X  4:  —  21Q   cu.  ft. 


18!^.  How  do  you  find  the  cubic  measure  of  a  volume  or  solid? 


1st 

OPEIlxVTION. 

8  ft. 

9' 

5" 

3 

6' 

4 

4' 

8"     6'" 

26 

4 

3 

MULTIPLICATION.  159 

Bnle. — Multiply  the  length,  breadth,  and  thickness  together. 

183.    When  the  Dimensions   are  in  feet  and  12ths  of  a  foot. 

Multiply  8  ft.  9'  5"  by  3  ft.  6',  and  theu  the  product  by 
2  ft.  6'. 

Analysis.— First  multiply    8    ft.  9' 
5"  by  6'.     Since  5"  =y^j  ft.,  and  6'  = 

ft,  or  30  thirds.  Since  12"'  =  1", 
30'"  ^12  =  2"  and  G'"  over,  which 
write  down. 

Then  9'  x  G'  =  y"^  x  ii  =  ui  sq.  ft., 
or    54",    to  which  add   the   2"   found 
ill  the  last  product,  making  56".    Then,         30  sq.ft.  8'      11"     6'" 
since  12"  =  1',  56"  -M2  =  4'  and   8" 
over,  which  write  down. 

Then  8  fectxG'  =  8  ft.  x  -^.j  ft.  =  *§  sq.  ft,  to  wliidi  add 
!ie  4'  from  the  last  product,  makuig  52'.  Then,  since  12'  =  1 
ijuare    foot,  52'  H-  12  =  4  sq.  ft*  and  4',  both  of  which  set  down. 

We  next  multiply,  in  the  same  manner,  by  3  feet,  giving  a 
product  of  2G  sq.  ft  4'  3".  The  sum  of  the  partial  products,  30  sq. 
ft.  8'  11"  6'",  is  the  first  required  product 

Now,  nmltiply  by  2  ft.  6'. 
First,  G'"  X  C'  =  y^\e  sq.  ft  x  j%  ft 
:=  :.%,  cu.  ft  =  36""  cu.  ft  =3"'. 
[.^u  11"  X  6'  =  ,VV  sq.  ft  X  {^  ft 
f'U.  ft.  and  three  added  from 
o  liwt  product  gives  60'"  =  5"  and 
"  over,  which  write  down. 

riirli     S'   X   G'    =    ^%     sq.     ft.    X    -i'V  ft 

-=  ,V,-    cu.    ft  =  48'',    to    which  add       76cu.  ft.  10'      4"  9'' 
5"  from   the  last  product,  gives  53"  = 
4'   and  5"  over,  whicli  v.'rite  down. 

Then,  30  sq.  ft  x  -/'^  ft.  =  W  -  180',  to  whicli  add  4  from  the 
last  product,  making  184'  :=r  15  cu.  ft  and  4'.     Next,   multiply   by 


2d 

OrERATION. 

30 

sq. 

ft. 

8' 

11" 

6' 

rf 

2 

6' 

15 

4' 

5" 

9' 

ff 

61 

5 

11 

160  DUODECIMALS. 

2  feet,  giving  the  partial  product  61  cu.  ft.  5'  11";  and  the  sum 
76  cu.  ft.  10'  4"  9"'  is  the  entire  product,  in  cubic  feet  and  12ths  of 
cubic  feet. 

Rule. — I.  Place  the  multiplier  under  the  multiplicand,  so 
that  units  of  the  same  order  shall  fall  in  the  same  column : 

II.  Multiply  the  mxdtiplicand  by  each  term  of  the  multiplier 
in  succession,  beginning  with  the  lowest  unit  of  each,  and 
make  the  index  of  each  product  equal  to  the  sum  of  the  in- 
dices of  the  factors : 

III.  Reduce  each  product^  as  it  arises,  to  the  next  higher 
unit;  write  down  the  remainder,  and  carry  the  quotient  to 
the  next  product : 

ly.   Find  the  sum  of  the  several  products. 

Examples. 

1.  How  many  cubic  feet  in  a  stick  of  timber  12  feet  6 
inches  long,  1  foot  5  inches  broad,  and  2  feet  4  inches 
thick  ? 

2.  Multiply  9  ft.  6'  by  4  ft.  V. 

3.  Multiply  12  ft.  5'  by  6  ft.  8'. 

4.  Multiply  35  ft.  4'  6"  by  9  ft.  10'. 

5.  What  is  the  product  of  45  ft.  4'  3"  by  12  ft.  2'  9"  ? 

6.  What  is  the  product  of   140  ft.  0'  2"  4'"  by  20  ft.  10'? 
t.   What  is  the  product  of  219  ft.  10'  6"  by  8'  4"  ? 

8.  What  are  the  contents  of  a  board  14  ft.  6'  3"  long,  and 
2  ft.  9'  wide  ? 

9.  How  many  square  feet  in  a  floor  18  ft.  9'  long,  and  15  ft. 
10'  wide? 

10.  How  many  square  yards  in  a  ceiling  tO  ft.  9'  long,  and 
12  ft.  3'  wide  ? 

11.  How  many  square  feet  are  there  in  a  ceiling  whose 
ength   is    15   feet,   and   width   42   feet  ? 

12.  How  many  square  yards  are  there  in  a  lot  of  ground 
wliose   length   is    118  feet,   and   width   25   feet? 

13.  How  many  square  feet  are  there  in  a  board  whose 
length   is    18    feet,   and   breadth    14   inches? 


MULTIPLICATION.  161 

14.  What  is  the  cost  of  painting  the  side  of  a  house  that  is  2T 
feet  high  and  22  feet  wide,  at  40  cents  per  square  yard? 

15.  How  many  acres  are  there  in  a  field  whose  length  is 
45  rods,  and  width  3Y  rods? 

16.  What  is  the  area  of  a  piece  of  ground  that  is  112  ft. 

5  in.  long,  and  27  ft.  9  in.  wide  ? 

17.  How  many  flagstones,  that  are  4ft.  Gin.  by  4ft., 
will  be  required  to  cover  a  walk  which  is  6  ft.  9  in.  wide  and 
2G4ft.  long? 

18.  What  will  be  the  cost  of  paving  a  yard  64  ft.  6' 
square,  at  5  cents  a  square  foot  ? 

19.  What  are  the  cubic  contents  of  a  block  of  marble  6  ft. 
9'  long,   4ft.   8'  wide,  and   2ft.  10'  thick? 

20.  There  is  a  room  97  feet  4'  around  it ;  it  is  9  feet  6' 
high:  what  will  it  cost  to  paint  the  walls,  at  18  cents  a 
square  yard  ? 

21.  What*  is  the  cubic  measure  of  a  pile  of  wood  that  is 
18  ft.  long,  7  ft.  high,  and  4  ft.  wide  ? 

22.  How  many  cords  are  there  in  a  pile  of  wood  that  is 
48  ft.  long,  9  ft.  high,  and  3  ft.  6  in.  wide  ? 

23.  A  gallon  contains  231  cubic  inches  :  how  many  gallons 
of  air  are  contained  in  a  room,  which  is  21  ft.  6  in.  long,  15  ft. 
wide,  and  10  ft.  high? 

24.  A  common  brick  is  8  in.  long,  4  in.  wide,  and  2  in. 
thick  :  how  many  bricks  are  there  in  a  pile,  whose  height  is 
12  ft.  4  in.,  width  8ft.,  and  length  15  ft.  9  in.,  supposing  no 
waste  space  ? 

25.  A  ditch  surrounds  a  plot  of  ground  which  is  240  ft. 
long,  and  164  ft.  wide.  The  ditch  is  3  ft.  6  in.  wide,  and  6  ft. 
9  in.  deep.     What  is  the  cubic  measure  of  the  ditch  ? 

26.  How  many  cubic  feet   of  wood  in  a  pile  36  ft.  5'  long, 

6  ft.  8'  high,  and  3  ft.  6'  wide? 

27.  What  will  a  pile  of  wood  26ft.  8'  long,  6ft.  Gin. 
high,  and  3ft.  3'  wide  cost  at  $3.50  a  cord? 

28     How  many  cu])ic  yards  of  earth  were  dug  from  a  cellar 


162  DUODECIMALS. 

which  measured  38  ft.  10'  long,  20  ft.  6'  wide,  and  9  ft.  4' 
deep  ? 

29.  x\t  16  cents  a  yard,  what  will  it  cost  to  plaster  a  room 
22  ft.  8'  long,  18  ft.  9'  wide,  and  11  ft.  6'  high?  There  are 
to  be  deducted  8  windows,  each  6  ft.  4'  high,  and  2  ft.  9'  wide  ; 
2  doors,  each  7  ft.  6'  high,  and  3  ft.  2'  widej  and  the  base 
moulding,  which  is  1  foot  wide? 

DIVISION. 

184.  Division  of  Duodecimals  is  the  operation  of  finding 
from  two  duodecimal  numbers  a  third,  which  multiplied  by  the 
first,  will  give  the  second? 

I.  The  floor  of  a  hall  contains  103 sq.  ft.  4'  5"  8'"  4'',  and 
is  6ft.  11'  8"  wide  :  what  is  its  length? 

Analysis.  —  The  operation. 

anits  of  the  dividend        ft.  sq.  ft.  ft. 

are  square  feet  and  6  11'  8")  103  4'  5"  8'"  4*^(14  9'  11" 
fractions  of  a  square  9t   *7'  4" 

foot.     The   units   of  c    q/   i /r  Q^fti 

the  divisor  are  linear  ^,     „     ,„ 

feet  and  fractions  of  - 

a  linear  foot  6'  4"  8'"  4" 

First,  consider  how        '  6'  4"  8'"  4'^ 

often  the    first    two 

parts  of  the  divisor  are  contained  in  the  first  part  of  the  dividend. 
The  first  two  parts  of  the  divisor  are  nearly  equal  to  7  feet,  and 
this  is  contained  in  103  sq.  ft.  14  times  and  something  over. 
Multiplying  the  divisor  by  this  term  of  the  quotient  and  subtract- 
ing, we  find  the  remainder  5  ft.  9'  1",  to  which  bring  down  8'". 

Next,  consider  how  many  times  the  first  two  parts  of  the  divisor, 
ecjuai  to  7  feet  nearly)  are  contained  in  the  first  two  parts  of  the 
remainder,  reduced  to  the  next  lower  unit;  that  is  5  ft.  9' =  CO'. 
Multiplying  the  divisor  by  the  quotient  figure  9',  and  making  the 
-ubtraction,  we  have  6'  4"  8",  to  whicli  bring  down  4". 


184.   Wliat  is  the  division  of  duodecimals?    How  is  it  performed? 


I 


DUODECIMALS.  163 

Consider,  again,  how  often,  nearly^  7  feet  is  ^contained  in  6'  4" 
:z^  70".  Multiplying  the  divisor  bj  the  quotient  11",  we  find  a 
[)roduct  equal  to  tlie  last  remainder.     Hence, 

The  process  of  division  is  the  same  as  that  in  other  de- 
nominate numbers,  except  in  the  manner  of  selecting  the 
quotient  figure. 

185.     Principles  of  the  operation. 

Notes. — 1  If  the  integral  unit  of  the  dividend  and  divisor  is  ilu- 
same,  the  unit  of  tJie  quotient  will  le  abstract. 

2.  If  the  unit  of  the  dividend  is  a  superficial  unit,  and  the  unit 
of  the  divisor  a  linear  unit,  the  unit  of  the  quotient  will  be  liintu 

S.   If  the  unit  of  dividend  is  a  unit  of  volume,  and  the  unit  ol    ' 
divisor  linear,  the  unit  of  the  quotient  mil  be  superficial. 

4.    If  the  unit  of  the  dividend  is  a  unit  of  volume,  and  tlu-  i. 
the  divisor  superficial,  the  ludt  of  the  quotient  wUl  be  linear. 

Examples.  * 

1     D.vido  29  sq.  ft.  0'  4"  by  6  ft.  4'. 
_.    Divide  50  sq.  ft.  0'  10"  6'"  by  9  ft.  6'. 
:].    Wiittt   is    the  length  of  a  floor  whose  area  is   iUoxp  ti 
1     (>",  and  breadth  24  ft.  3'? 

4.  A  load  of  wood,  contaiuiug  119cu.ft.  2'  G"  8'",  is  3  ft. 
4'  high,  and  4  ft.  2'  wide:  what  is  its  length? 

5.  In  a  granite  pillar  there  are  105  cu.  ft.  5'  V  (j'" ;  it 
is  3  ft.  9'    wide,  and  2  ft.  3'  thick  :  what  is  its  length  ? 

6.  There  are  394  sq.  ft.  2'  9''  in  the  floor  of  a  hall  that 
is  10  ft.  7'  wide:  what  is  its  length? 

7.  A  board  17  ft.  6'  long,  contains  27  sq.  ft.  8'  6'':  what 
is  its  width? 

8.  From  a  cellar  42  ft.  10'  long,  12  ft.  6'  wide,  were  thrown 
158  cu.  yd.  17  cu.  ft.  4'  of  earth:  how  deep  was  it? 

9.  A  block  of  marble  contains  86  cu.  ft.  2'  7"  9'"  6".  It 
is  4ft.  8'  wide  and  2ft.   10'  thick:  what  is  its  length? 


185.  What  are  the  principles  of  the  operation  ? 


164  DECIMAL    FKACTIONS, 


DECIMAL    FRACTIONS. 

186.  There  are  two  kiuds  of  Fractions  in  general  use  :  Com- 
mon Fractions  and  Decimal  Fractions. 

A  Common  Fraction  is  one  whose  unit  is  divided  into  any 
number  of  equal  parts. 

A  Decimal  Fraction  is  one  whose  unit  is  divided  according 
to  the  scale  of  tens. 

187.  If  the  unit  1  be  divided  into  10  equal  parts,  each  part 
Is  called  one-tenth. 

If  the  unit  1  be  divided  into  one  hundred  equal  parts,  or 
each  tenth  into  ten  equal  parts,  each  part  is  called  one-hun- 
dredth. 

If  the  unit  1  be  divided  into  one  thousand  equal  parts,  or 
each  hundredth  into  ten  equal  parts,  the  parts  are  called  thou- 
sandths, and  we  have  like  expressions  for  the  parts,  when  the 
unit  is  further  divided  accordinir  to  the  scale  of  tens. 


'O 


These  fractions  may  be  wri 

Three-tenths, 
Seventh-tenths,  - 
Sixty-five  hundredths, 
215  thousandths, 
1275  ten-thousandths, 


ten  thus 


t'o- 
tV- 

100' 

1000- 
1275 

loooo- 


From  which  we  see,  that  the  fractional  unit  of  a  decimal  is 
one  of  the  equal  parts  arising  from  dividing  the  unit  1  accord- 
ing to  the  scale  of  tens  :  hence,  it  is  one-tenth,  one-hundredth, 
"ne-thousandth,  &c. 

188.  A  Decimal  Number,  or  decimal,  is  one  which  contains 
a  decimal  unit. 

189.  A  Mixed  Decimal,  is  one  composed  of  a  whole  num- 
ber and  a  decimal. 


NOTATION  AND   NUMERATION.  165 


Notation  and  Numeration. 

190.  Tlae  denominators  of  decimal  fractions  are  seldom 
written.  The  fractions  are  expressed  by  means  of  a  period, 
placed  at  the  left  of  the  numerator,  called  the  decimal  point  (.). 

Thus,        y^o        .is  written     .3 

Too    -"^ 

2  15  II        OIK 

TodS -^^^ 

^i-2JL5_            ...     "     1275 
Toooo .A^»u 

The  denominator,  however,  of  every  decimal,  is  always  un- 
derstood : 

It  is  the  unit  1,  with  as  many  ciphers  annexed  as  there 
are  places  of  figures  in  the  decimal. 

The  place  next  to  the  decimal  point  is  called  the  place  of 
tenths,  and  its  unit  is  1  tenth  ;  the  next  place,  at  the  right, 
is  the  place  of  hundredths,  and  its  unit  is  1  hundredth  ;  the 
next  is  the  place  of  thousandths,  and  its  unit  is  1  thousandth  ; 
and  similarly  for  places  still  to  the  right. 

Decimal  Numeration  Table. 

(A 

;5    =0     '     <n    o 


•^     'S        3     rd     ^     3 

«  »i  S  ^  "  c  fl 
a  ^  o  a  g  S  a 
EH  K  H  H  W  3  H 

.4 is  read  4  tenths. 

,54 "54  hundredths. 

.064....  "  C4  thousandths. 

.6754...  "  6754  ten-thousandths. 

.01234..  "  1234  hundred-thousandths. 

.007654.  "  7654  millionths. 

.0043604  "  43604  ten-miUionths. 


166  DECIMALS. 

Note. — Decimal  fractions  are  numerated  from  left  to  right;  tlius, 
tenths,  TiundredtTis,  thousandths,  &c.  They  are  read  4  tenths,  54  hun 
dredths,  G4  thousandths,  &c. 

Whole  numbers  and  decimals  written  together. 


Whole  numbers. 

Decimals. 

4 

a 

^ 

g  «• 

^ 

o3 

1 

-i 

2     >-      S    r^      ^      i-     fl 

•5  'cJ    a  -V  "^    2    1 
1  g   2  S   gS  « 

a 

illions. 
undreds  o 
jns  of  tho 
lousands. 
undreds. 

ii 

'S 

42031451.2043018 


191.     Principles. 

1.  That  the  denominator  belonging  to  any  decimal  fraction  is  1, 
with  as  many  ciphers  annexed  as  there  are  places  of  figures  in  the 
decimal. 

2.  That  the  unit  of  any  place  is  ten  times  as  great  as  the  unit  of 
the  next  place  to  the  right — the  same  as  in  whole  numbers:  hence, 
whole  numbers  and  decimals  may  be  wri^jtcn  together,  by  placing  the 
decimal  point  between  them. 

186.  How  many  kinds  of  fractions  are  there?  What  are  they? 
What  is  a  common  fraction  ?  What  is  a  decimal  fraction  ? — 187.  Wlien 
the  unit  1  is  divided  into  10  equal  parts,  what  is  each  part  called? 
What  is  each  part  called  when  it  is  divided  into  100  equal  pans? 
When  into  1000?  Into  10,000?  &c.  How  are  decimal  fractions 
formed? — 188.  What  is  a  decimal  number? — 189.  What  is  a  mixed 
decimal? — 190.  Are  the  denominators  of  decimal  fractions  generally 
set  down  ?  How  are  the  fractions  expressed  ?  Is  the  denominator 
understood  ?  What  is  it  ?  What  is  the  place  next  the  decimal  point 
called?  What  is  its  unit?  What  is  the  next  place  called?  What  is 
its  unit  ?  What  is  the  third  place  called  ?  What  is  its  unit  ?  Which 
way  are  decimals  numerated  and  read? — 191.  What  are  the  two  prin 
ciples  of  decimals? 


NOTATION    AND    NUMERATION.  167 

192.  Rule  for  Writing  Decimals. 

Write  the  decimal  as  if  it  icere  a  ichole  number,  prefixing 
as  many  ciphers  as  are  necessary  to  make  its  right-hand  Jig- 
lire  of  the  required  name. 

193.  Rule  for  Reading  Decimals. 

Bead  the  decimal  as  though  it  were  a  whole  number^ 
adding  the  denomination  indicated  by  the  lowest  decimal  unit. 

Examples. 
Write  the  following  common  fractions  decimally : 

(1.)  (2.)  (3.)  (4.)  (5.) 

6  17  5  2  7  47 

ToT  Tft  TBoT  Toll  TinFS- 


(6.)  (t.)  (8.)  (9.)  (10.) 

6AV  lioVs  9rfi»  lOm  12-W. 

Write  the  following  numbers  in  figures,  and  numerate  them  : 

1.  Twenty-seven,  and  four-tenths. 

2.  Thirty-six,  and  fifteen-thousandths. 

3.  Ninety-nine,  and  twenty-seven  ten-thousandths. 

4.  Three  hundred  and  twenty  thousandths. 

5.  Two  hundred,  and  three  hundred  and  twenty  millionths. 

6.  Three  thousand  six  hundred  ten-thousandths. 
T.   Five,  and  three-millionths. 

8.  Forty,  and  nine  ten-millionths. 

9.  Forty-nine  hundred  ten-thousandths. 

10.  Fifty-nine,  and  sixty-seven  ten-thousandths. 

11.  Four  hundred  and  sixty-nine  ten-thousandths. 

12.  Seventy-nine,  and  four  hundred  and  fifteen  millionths. 

13.  Sixty-seven,    and    two    hundred    and    2t    ten-lOOOths. 

14.  One  hundred  and  five,  and  ninety-five  ten-millionths. 

15.  Forty,  and  204  thousand  millionths. 

192.  What  is   the  rule  for  writing  decimals  ?— 193.  What  is  the 
rule  for  reading  decimals? 


168  DECIMALS. 


UNITED    STATES    MONEY. 

194.  The  denominations  of  United  States  Money  correspond 
to  the  decimal  division,  if  we  regard   one   dollar   as  tJie  unit  : 

For,  the  dimes  are  tenths  of  the  dollar,  the  cents  are  him 
dredths  of  the  dollar,  and  the  mills,  being  tenths  of  a  cent, 
are  thousandths  of  the  dollar. 

Examples. 

1.  Express  $31  and  26  cents  and  5  mills,  decimally. 

2.  Express  $lt  and  5  mills,  decimally. 

3.  Express  $215  and  8  cents,  decimally. 

4.  Express  $2t5  5  mills,  decimally. 

5.  Express  $9  8  mills,  decimally. 

6.  Express  $15  6  cents  9  mills,  decimally, 
t.  Express  $21  18  cents  2  mills,  decimally. 
8.  Express  $3  5  cents  9  mills,  decimally. 

ANNEXING    AND    PREFIXING    CIPHERS. 

195.  Annexing  a  cipher  is  placing  it  on  the  right  of  a 
number. 

If  a  cipher  is  annexed  to  a  decimal  it  makes  one  more  de- 
cimal place,  and  therefore,  a  cipher  must  also  be  added  to 
the  denominator  (Art.  190). 

The   numerator   and   denominator   will   therefore    have    been 


194.  If  tlie  denominations  of  Federal  Money  be  expressed  decimally, 
what  is  tlie  miit?  What  part  of  a  dollar  is  1  dime?  What  part  Oi 
a  dime  is  1  cent?  What  part  of  a  cent  is  a  mill?  What  part  of  a 
dollar  is  1  cent?  1  mill? — 195.  Wlien  is  a  cipher  annexed  to  a  num- 
ber ?  Does  the  annexing  of  ciphers  to  a  decimal  alter  its  value  ?  Why 
not?  What  does  five-tenths  become  by  annexing  a  cipher?  What 
by  annexing  two  ciphers?    T'iree  ciphers? 


DKCIMAL    FUACTIONS.  1G9 

multiplied  by  the  same  number,  and  consequently   the  value  of 
the  fraction  will  not  be  changed  (Art.  143)  :  hence, 

Annexing  ciphers  to  a  decimal  does  not  alter  its  value. 

Take  as  an  example,  .5  =  ^o. 

If  we  annex  a  cipher  to  the  decimal,  wo  at  the  same  time 
aimex  one  to  the  denominator ;   thus, 

.5  becomes      .50     =     ^^q      by  annexing  one  cipher. 
.5  becomes     .500    =     -f^Q     by  annexing  two  ciphers. 
.5  becomes  .5000    =     to°oVo  ^J  annexing  three  ciphers. 

196.   Prefixing  a  cipher  is  placing  it  on  the  left  of  a  number. 

If  ciphers  are  prefixed  to  a  decimal,  the  same  number 
of  ciphers  must  be  annexed  to  the  denominator ;  for,  the  de- 
nominator must  always  contain  as  many  ciphers  as  there  are 
decimal  places  in  the  numerator.  Now,  the  numerator  will  re- 
main unchanged  while  the  denominator  will  be  increased  ten 
times  for  every  cipher  annexed  ;  and  hence  the  value  of  the 
fraction  will  be  diminished  ten  times  for  every  cipher  pre- 
fixed to  the  decimal  (Art.  142)  :  hence. 

Prefixing  ciphers  to  a  decimal  diminishes  its  value  ten 
times  for  every  cipher  prefixed. 

Take,  for  example,  the  decimal  .3  =  ^. 
.3  becomes      .03     =     y^^      by  prefixing  one  cipher ; 
.3  becomes    .003     =     y^o'Q    ^7  prefixing  two  ciphers ; 
.3  becomes  .0003     =     yo^irff  ^7  prefixing  three  ciphers  : 

in  which  the  fraction  is  diminished  ten   times  for  every  cipher 
prefixed. 

196.  When  is  a  cipher  prefixed  to  a  number  ?  When  prefixed  to  a 
decimal,  does  it  increase  the  numerator  ?  Does  it  increase  the  denom- 
inator?   What  effect,  then,  has  it  on  the  value  of  the  decimal? 

8 


170  ADDITION   OF 

197.     Analysis  of  decimals. 

Analyze  62.25.  It  is  composed  of  6  tens,  2  units,  2  tenths, 
and  6  hundredths  ;  or  it  is  composed  of  62  units  and  25  hun- 
dredths ;  or  of  622  tenths  and  5  hundredths  ;  or  6225  hun- 
dredths. 

Note.— Let  it  be  remembered  that  a  fractional  unit  of  any  one 
place  is  y\j  of  the  unit  of  the  place  next  on  the  left,  or  yl^  of  the 
miit  wliich  is  2  places  to  the  left,  or  y^Vff  ^^  ^^^  fractional  unit 
which  is  three  places  to  the  left. 

ADDITION    OF    DECIMALS. 
198.   Addition  of  Decimals  is   the   operation  of  finding  the 
sum  of  two  or  more  decunal  numbers. 

It  must  be  remembered,  that  only  units  of  the  same  value 
can  be  added  together.  Therefore,  in  setting  down  decimal 
numbers  for  addition,  figures  having  the  same  unit  value  must 
be  placed  in  the  same  column. 

The  addition  of  decimals  is  then  made  in  the  same  manner 
as  that  of  whole  numbers. 

1.   Find  the  sum  of  87.06,  327.3,  and  .0567. 

OPKRATION. 

Analysis. — Place  the  decimal  points  in  the  same  87  06 

column:  this  brings  units  of  the  same  value  in  the  qo>7  q 

same    column:    then    add    as    in    whole    numbers: 
hence, 


.0567 


Rule. 


414.4167 


I.   Set  down   the  numbers  to  he  added  so  that  figures  oj 
the  same  unit  value  shall  stand  in  the  same  column : 


198.  What  is  addition  ?  What  parts  of  a  unit  may  be  added  to- 
gether? How  do  you  set  down  the  numbers  for  addition?  How  will 
the  decimal  points  fall  ?  How  do  you  then  add  ?  How  many  decimal 
places  do  you  point  off  in  the  sum  ? 


DKCIMALS.  171 

II.   Add  as  in  simple  numbers,  and  point  off  in  the  sum, 

from  the  right  hand,  a  number  of  2)laces  for  decimals  equal 
to  the  greatest  number  of  places  in  any  of  the  numbers 
added. 

Proof. — The  same  as  in  simple  numbers. 


Examples. 

1.  Add  G.035,  763.196,  445.3741,  and  91.5754   together 

2.  Add  465.103113,  .78012,  1.34976,  .3549,  and  61.11. 

3.  Add  57.406  +  97.004  +  4  +  .6  +  .06  +  .3. 

4.  Add  .0009  +  1.0436  +  .4  +  .05  +  .047. 

5.  Add  .0049  +  49.0426  +  37.0410  +  360.0039. 

6.  Add  5.714,  3.456,  .543,  17.4957  together. 

7.  Add  3.754,  47.5,  .00857,  37.5  together. 

8.  Add  54.34,  .375,  14.795,  1.5  together. 

9.  Add  71.25,  1.749,  1759.5,  3.1  together. 

10.  Add  375.94,  5.732,  14.375,  1.5  together. 

11.  Add  .005,  .0057,  31.008,  .00594  together. 

12.  Required  the  sum  of  9  tens,  19  hundredths,  18  thou- 
sandths, 211  hundred-thousandths,  and  19  millionths. 

13.  Find  the  sum  of  two,  and  twenty-five  thousandths,  five 
and  twenty-seven  ten-thousandths,  forty-seven,  and  one  hun- 
dred twenty-six  millionths,  one  hundred  fifty,  and  seventeen  ten- 
millionths. 

14.  Find  the  sum  of  three  hundred  twenty-seven  thousandths, 
fifty-six  ten-thousandths,  four  hundred,  eighty-four  millionths, 
and  one  thousand  five  hundred  sixty  hundred-millionths. 

If).   What  is  the  sum  of  .5  hundredths,  27  thousandths,  476 
ladred-thousandths,    190    ten-thousandths,    and    1*279    ten-mil- 
lionths  ? 

16.  What  is  the  sum  of  25  dollars  12  cents  6  mills,  9  dol- 


172  DECIMALS. 

lars  8  cents,  12  dollars  t  dimes  4  cents,  18  dollars  5  dimes  8 
mills,  and  20  dollars  9  mills  ? 

It.  What  is  the  sum  of  126  dollars  9  dimes,  420  dollars 
15  cents  6  mills,  317  dollars  6  cents  1  mill,  and  200  dollars 
4  dimes  1  cents  3  mills? 

18.  A  man  bought  4  loads  of  hay,  the  first  contained  1  ton 
25  thousandths  ;  the  second,  99 1  thousandths  of  a  ton  ;  the 
thu'd,  88  hundredths  of  a  ton ;  and  the  fourth,  9876  ten- 
thousandths  of  a  ton  :  what  was  the  entire  weight  of  the  four 
loads  ? 

19.  Paid  for  a  span  of  horses,  $225.50  ;  for  a  carriage, 
$127,055  ;  and  for  harness  and  robes,  $75.28  :  what  was  the 
entire  cost  ? 

20.  Bought  a  barrel  of  flour,  for  $9,375  ;  a  cord  of  wood, 
for  $2.12J  ;  a  barrelof  apples,  for  $1.62J  ;  and  a  quarter  of 
beef,  for  $6.09  :  what  was  the  amount  of  my  bill  ? 

21.  A  farmer  sold  grain  as  follows  :  wheat,  for  $296.75  ; 
corn,  for  $126.12^;  oats,  for  $97.37J  ;  rye,  for  $100.10;  and 
barley,  for  $50.62^  :  what  was  the  amount  of  his  sale  ? 

22.  A  person  made  the  following  bill  at  a  store  :  5  yards 
of  cloth,  for  $16,408;  2  hats,  for  $4.87};  4  pairs  of  shoes, 
for  $6  ;  20  yards  of  calico,  for  $2,378  ;  and  12  skeins  of  silk, 
for  $0.62 J  :  what  was  the  amount  of  his  bill  ? 

23.  What  is  the  sum  of. $99  87  cents  5  mills;  $87  6  cents 
18  mills  ;  $59  42  cents  20  mills  ;  $60  49  cents  16  mills  ;  and 
$21  29  cents  13  mills? 

24.  What  is  the  sum  of  $97  4  mills  ;  $25  19  mills  ;  $65  95 
cents  6  mills  ;  $4  87J  cents  3  mills  ;  and  $55  14}  cents  9 
mills  ? 

25.  Mr.  James  bought  of  Mr.  Squires,  the  grocer,  the  fol 
lowing  articles  :  a  bag  of  coffee,  for  $37,874  ;  a  chest  of  tea, 
for  $50,009  ;  a  barrel  of  sugar,  for  $19  4  cents  and  6  mills  ; 
and  9  gallons  of  wine,  for  $27  69  cents  and  15  mills  :  what 
was  the  amount  of  his  1)111  ? 


DECIMALS.  178 


SUBTRACTION 

199.    Subtraction  of    Decimals    is    the   operation  of  finding 
tlic  difference  between  two  decimal    numbers. 

1.    From  6.304  take  .0563. 

Analysis. — In  this  example  a  cipher  is  annexed  orEiiATioN. 

o   the  minuend  to  make   the  number  of  decimal  n  ^040 

places    equal    to    the    number   in    the    subtrahend.  0563 

This    does    not    alter  the   value    of   the    minuend         

(Art.  195):   hence,  6.2477 

Rule. 

I.  Write  the  less  number  under  the  greater,  so  that  figures 
of  the  same  mat  value  shall  fall  in  the  same  column: 

II.  Subtract  as  in  simjjle  numbers,  and  point  off  the  deci- 
mal jilaces  in   the   remainder,  as  in  addition. 

PnooF. — Same  as  in  simple  numbers. 


Examples. 

1.  From  875.05  take  .0467. 

2.  From  410.0591   take  41.496. 

3.  From  7141.604  take  .09046. 

4.  Required  the  difference  between  57.49  and  5.768. 
6.  What  is  the  difference  between  .3054  and  3.075  ? 

6.  Required  the  difference  between  1745.3  and  173.45. 

7.  What  is  the  difference  between  seven-tenths   and   54   ten- 
thousandths  ? 

8.  What  is  the  difference  between  .105  and  1.00075  ? 

9.  What  is  the  difference  between  150.43  and  754.365  ? 

10.  From  1754.754  take  375.49478. 


L 


199.  What  is  subtraction  of  decimal  fractions?  How  do  you  set 
down  tlie  numbers  for  subtraction  ?  How  do  you  then  subtract  1 
How  many  decimal  places  do  you  point  off  in  the  remainder  ?  What 
js  the  proof? 


Vti  DECIMALS. 

11.  Take  ^5.304  from  175.01. 

12.  Required  the  difference  between  17.541  and  35.49. 

13.  Required  the  difference  between  t  tenths  and  7  mil- 
lionths. 

14.  From  396  take  67  and  8  ten-thousandths. 

15.  From  1  take  one-thousandth. 

16.  From  6374  take  fiftj-nine  and  one-tenth. 

17.  From  365.0075  take  5  millionths. 

18.  From  21.004  take  98  ten-thousandths. 

19.  From  260.3609  take  47  ten-millionths. 

20.  From  10.0302  take  19  millionths. 

21.  From  2.03  take  6  ten-thousandths. 

22.  From  one  thousand,  take   one-thousandth. 

23.  From   twenty-five   hundred,  take   twenty-five  hundredths. 

24.  From  two  hundred,  and  twenty-seven  thousandths,  take 
ninety-seven,  and  one  hundred  twenty  ten-thousandths. 

25.  A  man  owning  a  vessel,  sold  five  thousand  seven  hun- 
dred sixty-eight  ten-thousandths  of  her :  how  much  had  he 
left? 

26.  A  farmer  bought  at  one  time  127.25  acres  of  land  ;  at 
another,  84.125  acres  ;  at  another,  116.7  acres.  He  wishes  to 
make  his  farm  amount  to  500  acres  :  how  much  more  must  he 
purchase  ? 

27.  Bought  a  quantity  of  lumber  for  $617. 37J,  and  sold 
it  for   $700  :   how  much  did  I  gain  by  the  sale  ? 

28.  Having  bought  some  cattle  for  $325.50  ;  some  sheep 
for  $97.12i;  and  some  hogs  for  $60.87|  ;  I  sold  the  whole 
for  $510.10:   what  was  my  entire  gain? 

29.  A  dealer  in  coal  bought  225.025  tons  of  coal  :  ho 
sold  to  A,  1.05  tons  ;  to  B,  20.007  tons  ;  to  C,  40.1255  tons  -. 
and  to  D,   37.00056  tons  :    how  much  had  he  left  ? 

30.  A  man  owes  $2346.865:  and  has  due  him,  from  A, 
$1240.06 ;  and  from  B,  $1867.984  :  how  much  will  he  have 
left  after  paying  his  debts? 


DECIMALS.  175 

MULTIPLICATION. 

200.  Multiplication  of  decimals  is  the  operation  of  taking 
one  number  as  many  times  as  there  are  units  in  another,  when 
one  or  both  of  the  factors  contain  decimals. 

1.  Multiply  8.03  by  6.102. 

Analysis. — If  we  change 
both  factors  to  common  frac- 
tions, tlie  product  of  the  nu- 
merators will  be  the  same  as 
that  of  the  decimal  numbers, 
and  the  number  of  decimal 
places    will     he    equal   to    the 


OrERATION. 

8.03 

= 

StJij 

=  m 

6.102 

= 

6lW<F 

=  mi 

803 
TOO 

X 

fje? 

8.03 
6.102 

1606 
803 
number   of  ciphers  in   the  two  4818 

denomiTiatore ;  hence,  — 

48.99906 

Rule. — Multiply  as  in  sim2?le  nmnbei'S,  and  point  ojf  in 
the  2^roduct,  from  ike  rigid  hand,  as  many  figures  for  deci- 
mals as  there  are  decimal  places  in  both  faciors  ;  and  if  there 
he  not  so  many  in  the  product,  supply  the  deficiency  by  prefiX' 
ing  ciphers. 

Proof. — The  same  as  in  whole  numbers. 

Examples. 

1.  Multiply  2.125  by  375  thousandths. 

2.  Multiply  .4712  by  5  and  6  tenths. 

3.  Multiply  .0125  by  4  thousandths. 

4.  Multiply  6.002  by   25  hundredths. 
6.  Multiply  473.54  by  57  thousandths. 

6.  Multiply  137.549  by  75  and  437  thousandths. 

7.  Multiply  3,  .7495,  and  73487,  together. 

200,  What  is  multiplication  of  decimals?  After  multiplying,  liow 
many  decimal  places  -wiU  you  point  oflf  in  the  product  ?  When  there 
are  not  so  many  in  the  product,  what  do  you  do?  Give  the  rule  for 
the  multiplication  of  decimals. 


176  MULTIPLICATION   OF 

8.  Multiply  .043*15  by  illU  hundred-thousandths. 

9.  Multiply  .311343   by  seventy-five  thousand  493. 

10.  Multiply  49.0t54  by  3  and  5U4  ten-thousandths. 

11.  Multiply  .573005  by  T54  milKonths. 

12.  Multiply  .375494  by  574  and  375  hundredths. 

13.  Multiply  .000294  by  one  millionth. 

14.  Multiply  300.27  by  62. 

15.  Multiply  93.01401  by  10.03962. 

16.  Multiply  596.04  by  0.000012. 

17.  Multiply  38049.079  by  0.000016. 

18.  Multiply  1192.08  by  0.000024. 

19.  Multiply  76098.158  by   0.000032. 

20.  Multiply  thirty-six  thousand  by  thu'ty-six  thousandths. 

21.  Multiply  125  thousand  by  25  ten-thousandths. 

22.  Find  the  product  of  50  thousand  by  75  ten-millionths. 

23.  Find  the  product  of  48  hundredths  by  75  ten-thousandths. 

24.  What  are  the  contents  of  a  lot  of  land,  16.25  rods  long, 
and  9.125  rods  wide  ? 

25.  What  are  the  contents  of  a  board  12.07  feet  long,  and 
1.005  feet  wide? 

26.  What  will  27.5  yards  of  cloth  cost,  at  .875  dollars  per 
yard? 

37.   At  $25,125  an  acre,  what  will  127.045  acres  of  land  cost? 

28.  rBought  17.875  tons  of  hay,  at  $11.75  a  ton:  what  was 
the  cost  of  the  whole? 

29.  A  gentleman  purchased  a  farm  of  420.25  acres,  at 
135.08  an  acre;  he  afterwards  sold  196.175  acres  to  one  man 
at  ^37.50  an  acre,  and  the  remainder  to  another  person,  at 
$36,125  an  acre  :  what  did  he  gain  ? 

30.  A  merchant  bought  two  pieces  of  cloth,  one  containing 
37.5  yards,  at  $2.75  a  yard,  and  the  other,  containing  27.35 
yards,  at  $3,125  a  yard  ;  he  sold  the  whole  at  an  average 
price  of  $2.94  a  yard  :  did  he  gain  or  lose  by  the  bargain, 
and  how  much  ? 


100 


26749.6 


DECIMALS.  177 

CONTRACTIONS    IN    MULTIPLICATION. 

201.  Contractions,  in  the  multiplication  of  decimals,  are 
short    methods    of    finding  the   product 

CASE    I. 
202.    To  multiply  by  10,  100,  1000,  Ac. 
1.   Multiply  267.496  by  100. 

Analysis. — Removing  the  decimal  point  one  operation. 
place  to  the  right,  increases  the  value  of  the  deci-  267.496 
nial  ten  times;  removing  it  two  places,  one  hun- 
dred thues,  &c.  To  multiply  by  10,  100,  &c.,  we 
remove  the  decimal  point  as  many  places  to  the 
right  as  there  are  ciphers  in  the  multiplier:  hence, 

Rule. — Bemove  the  decimal  point  as  many  places  to  the 
right  as  there  are  ciphers  in  the  multiplier ;  annexing  ciphers, 
if  necessary. 

Examples. 

1.  Multiply  479.64  by  10  ;  also,  by  100. 

2.  Multiply  69.4729  by  1000  ;  also,  by  10. 

3.  Multiply  41.53  by  10000  ;  also,  by  100. 

4.  Multiply  27.04  by  100  ;  also,  by  1000. 

5.  Multiply  129.072  by  1000  ;  also,  by  10. 

6.  Multiply  87.1  by  10000  ;  also,  by  100. 

7.  Multiply  140.1  by  1000  ;  also,  by  10. 

CASE     II . 

203.  To  multiply  two  decimals,  and  retain  in  the  product  a 
certain  number  of  decimal  places. 

I.  Let  it  be  required  to  find  the  product  of  2.38645  mul 
tiplied  by  38.2175,  in  such  a  manner  that  it  shall  contain  but 
four  decimal  plrtces. 


17S  CONTRACTIONS. 

Analysis. — Write  the  unit  figure  of  the  mul-  operation. 

tiplier    under    that    place    of  the    multiplicand         2.38645 
which  has  the  same  number,  counted  from   the  38.2175 

decimal  point,  as  the  figures  to  be  retained  in  

the  product,  and  write  the  other  figures  in  their 

proper  places.     Kow,   the  product  of  the   unit         i^vjio 

figure  of  the  multiplier,  by  the  figure  of  the  mul-  ^  4773 

tiplicand  directly  over   it,   will  have   the   unit  239 

vahio  of  the  required  product.     The  product  of  167 

the  next  figure  at  the  right,  in  the  multiplicand,  12 

by  the  tens  of  the  multiplier,  will  also  give  a 

product   of   the   required   unit  value;    and  the 

same  will  be  true  for  the  product  of  any  two 

figures  equally  distant  from  the  unit  figure   of  the  multiplier  and 

the  figure  of  the  multiplicand  directly  over  it. 

In  regard  to  the  decimals,  we  observe,  that  the  tenths  multi- 
plied by  the  figure  at  the  left  of  the  one  standing  over  the  unit 
figure  of  the  multiplier,  will  give  a  product  of  the  required  unit 
value;  and  the  same  will  be  true  for  any  two  figures  equally  dis- 
tant from  the  decimal  point  and  from  the  figure  standing  over  the 
unit  place  of  the  multiplier. 

We  therefore  begin  the  operation  with  the  highest  unit  figure 
of  the  multiplier,  and  the  corresponding  figure  of  the  multiplicand, 
and  then  multiply  in  succession  by  the  figures  at  the  right.  We 
must  remember  that  the  whole  of  the  multijylicand  should  be  multi- 
plied by  every  figure  of  the  nuiltiplier.  Hence,  to  compensate  for 
the  parts  omitted,  we  begin  with  one  figure  to  the  right  of  that 
which  gives  the  true  unit,  and  carry  one  when  the  product  is 
greater  than  5  and  less  than  15 ;  2,  when  it  falls  between  15  and 
25;  8,  when  it  falls  between  25  and  35;  and  so  on  for  the  higher 
numbers. 

For  example,  when  we  multiply  by  the  8,  instead  of  saying 
8  times  4  are  82,  and  writing  down  the  2,  we  say  first,  8  times 
5  are  40,  and  then  carry  4  to  the  product  32,  which  gives  36 

201.  What  is  contraction  in  the  multiplication  of  decimals? — 202. 
How  do  you  multiply  by  10,  100,  &c?  If  there  are  not  as  many  decimal 
places  in  the  product  as  there  are  ciphers,  wliat  do  you  doV — 203.  Ex 
plain  the  manner  of  multiply hig  two  decimals  together  so  as  to  retaiu 
a  given  number  of  places  in  the  product. 


DECIMALS.  170 

So,  when  we  multiply  by  the  last  figure  5,  wo  first  say,  5  tunes 
8  are  15,  then  5  times  2  are  10  and  2  to  carry,  make  12,  wliicli 
is  written  down. 

Examples. 

1.   Multiply  3G.T463T   by   121.0463,  retaining  three  decimal 
laces  in  the  product. 


OONTRACnON. 

COMMON  WAY. 

36.74631 

36.74637 

127.0463 

127.0463 

3674637 

11023911 

734927 

22047822 

257225 

14698548 

1470 

25722459 

220 

7349274 

11 

3674637 

4668.490 

4668.490346931 

2.  Multiply  54.7494367  by  4.714753,  reserving  five  places  of 
decimals  in  the  product. 

3.  Multiply  475.710564  by  .3416494,  retaining  three  decimal 
places  in. the  product. 

4.  Multiply  3754.4078  by  .734576,  retaining  five  decimal 
places  in  the  product. 

6.  Multiply  4745.679  by  751,4549,  and  reserve  only  whole 
numbers  in  the  product. 

DIVISION. 

204.  Division  of  Decimals  is  the  operation  of  finding  how 
many  times  one  number  is  contained  in  another,  when  one  or 
both  are  decimals. 


204.  \Vliat  is  division  of  decimals?  How  is  division  performed? 
IIow  docs  the  number  of  decimal  places  in  the  dividend  compare  with 
those  in  the  divisor  and  quotient?  How  do  you  determine  tlie  num- 
ber of  deoimal  places  in  the  quotient?  Give  tlie  rule  for  the  division 
6t  decimals. 


180 


DIVISION    OF 


OPERATION. 

2.043).n505(35 
6129 

10215 
10215 

Ans.   0.35 


1.  Divide  the  decimal  .n505  by  2.043. 

Analysis. — Division  of  decimals  is  per- 
forme(^.  in  the  same  manner  as  division 
cf  whole  numbers.  Since  the  dividend 
must  be  equal  to  the  product  of  the  di- 
visor and  quotient,  it  must  contain  as 
many  decimal  places  as  both  of  them. 
(Art.  200.)    Therefore, 

There  must  he  as  many  decimal  places  in  the  quotient  as  the 
number  of  decimal  places  in  the  dividend  exceeds  that  in  the 
divisor:  Hence, 

Rule. — Divide  as  in  simple  numbers,  and  point  off  in  the 
quotient,  from  the  right  hand,  as  many  places  for  decimals 
as  the  number  of  decimal  places  in  the  dividend  exceeds  that 
in  the  divisor;  and  if  there  are  not  so  many,  supply  the 
deficiency  by  prefixing  ciphers. 


Exa.rn.pies. 


1.  Divide  4.6842  by  2.11. 

2.  Divide  12.82561  by  1.505. 

3.  Divide  33.66431  by  1.01. 

4.  Divide  .010001  by  .01. 

5.  Divide  24.8410  by  .002. 

6.  Divide  .0125  by  2.5. 


1.  Divide  .051  by  .012. 

8.  Divide  .063  by  9. 

9.  Divide  1.05  by  14. 

10.  Divide  5.1435  by  4.05. 

11.  Divide  .465^5  by  31.05. 

12.  Divide  2.46616  by  .145. 

By 


13.  What  is   the  quotient   of  75.15204,  divided   by  3? 
.3?     By  .03?     By  .003?     By  .0003. 

14.  What  is  the  quotient  of  389.27688,  divided  by  8  ?     By 
.08  ?     By  .008  ?     By  .0008  ?     By  .00008  ? 

15.  What  is   the  quotient   of  374.598,   divided   by  9  ?      By 
.9?     By  .09?     By  .009?     By  .0009?     By  .00009? 

16.  What  is    the  quotient   of  1528.4086488,  divided   by  6? 
By  .06?     By  .006?     .0006?     By  .00006  ?     By  .000006  ? 

n.  Divide  17.543275  by  125.7. 
18.  Divide  1437.5435  by  .7493. 


DECIMALS.  181 

19.  Divide  .000177089  by  .0374. 

20.  Divide  1674.35520  by  9.60. 

21.  Divide  120463.2000  by  1728. 

22.  Divide  47.54936  by  34.75. 

23.  Divide  74.35716  by  .00573. 

24.  Divide  .37545987  by  75.714. 

25.  If  25  men  remove  154.125  cubic  yards  of  earth  in  a 
day,  bow  much  does  each  man  remove? 

26.  If  167  dollars  8  dimes  7  cents  and  5  mills  be  equally 
divided  among  17  men,  how  much  will  each  receive? 

27.  Bought  45.22  yards  of  cloth  for  $97,223  :  how  much 
was  it  a  yard  ? 

28.  If  375.25  bushels  of  salt  cost  $232,655,  what  is  the 
price  per  bushel? 

29.  At  $0,125  per  pound,  how  much  sugar  can  be  bought 
for  $2.25  ? 

30.  How  many  suits  of  clothes  can  be  made  from  34  yards 
of  cloth,  allowing  4.25  yards  for  each  suit  ? 

31.  If  a  man  travel  26.18  miles  a  day,  how  long  will  it 
take  him  to  travel  366.52  miles  ? 

32.  A  miller  wishes  to  purchase  an  equal  quantity  of  wheat, 
corn,  and  rye  ;  he  pays  for  the  wheat,  $2,225  a  bushel  ;  for 
the  corn,  $0,985  a  bushel ;  and  for  the  rye,  $1,168  a  bushel : 
how  many  bushels  of  each  can  he  buy  for  $242,979  ? 

33.  A  farmer  purchased  a  farm  containing  56  acres  of  wood- 
land, for  which  he  paid  $46,347  per  acre  ;  176  acres  of  meadow 
laud,  at  the  rate  of  $59,465  per  acre  ;  besides  which  there  was 
a  swamp  on  the  farm  that  covered  37  acres,  for  which  he  was 

harged  $13,836  per  acre.     What  was  the   area  of  the   land  ; 
A^hat  its  cost ;  and  what  was  the  average  price  per  acre  ? 

34.  A  person  dying  has  $8345  in  cash,  and  6  houses,  valued 
at  $4379.837  each  ;  he  ordered  his  debts  to  be  paid,  amount- 
ing to  $3976.480,  and  $120  to  be  expended  at  his  funeral; 
the  residue  was  to  be  divided  among  his  five  sons   in   the  fol- 


182  DIVISION  OF 

lowing  manner :  the  eldest  was  to  have  a  fourth  part,  and 
each  of  the  other  sons  to  have  equal  shares.  What  was  the 
share  of  each  son  ? 

205.    When  the  decimal  places  of  the  divisor  exceed  those  of 
the  dividend. 

When  there  are  more  decimal  places  in  the  divisor  tha  ^ 
in  the  dividend,  annex  as  many  ciphers  to  the  dividend  as 
are  necessary  to  make  its  decimal  places  equal  to  those  of  the 
divisor  ;  all  the  figures  of  the  quotient  will  then  he  whole  num- 
bers. And  always  bear  in  mind  that,  the  number  of  decimal 
places  in  the  quotient^  is  equal  to  the  excess  of  the  number  in' 
the  dividend  over  the  number  in  the  divisor. 
Examples. 

L  Divide  439t.4  by  3.49.  operation. 

3,49)4397.40(l260^ns. 

349 

Analysis. — We    annex    one  0  to    the 

dividend.     Had  it  contained    no   decimal  "^^ 

place,  we  should  have  annexed  two.  ^98 

2094 
2094 


2.  Divide    1097.01097    by   .100001. 

3.  Divide  9811.0047  by  .1629735. 

4.  Divide  .1  by  one  ten-thousandth. 

5.  Divide  10  by  one-tenth. 

6.  Divide  6  by  .6.  By  .06,  By  .006.  By  .2.  By  .3.  By 
.003.     By  .5.     By  .005.     By  .000012. 

206.    When  the  division  does   not  terminate. 

When  it  is  necessary  to  continue  the  division  further  than 
the  figures  of  the  dividend  will  permit,  we  may  annex  ciphers 
to  it,  and  consider  them  as  decimal  places. 

205.  What  do  you  do  when  the  decimal  places  of  the  diviaor  ex- 
ceed those  of  the  dividend?    What  will  the  quotient  {hen  t)e! 


DECIMALS. 


183 


Examples. 

1.  Divide  4.25  by  1.25. 

Analysis. — In  this  example,  after  having 
exhausted  the  decimals  of  the  dividend,  we 
annex  a  0,  and  then  the  decimal  places 
used  in  the  dividend  will  exceed  those  in 
the  divisor  by  1. 

2.  Divide  .2' by  .06. 

Analysis, — We  see,  that  in  this  example, 
the  division  will  never  terminate.  In  such 
cases,  the  division  sliould  be  carried  to  the 
third  or  fourth  place,  which  will  give  the 
answer  true  enough  for  all  practical  pur- 
poses, and  the  sign  +  should  then  be  writ- 
ten, to  show  that  the  division  may  still  be 
continued. 


OPERATION. 

1.25)4.25(3.4 
3.75 
500 
500 

Ans.  3.4 

OPERATION. 

.06).20(3.333  + 
18 
20 
18 
20 
18 
2 
Ans,  3.333  -|- 


3.  Divide  37.4  by  4.5. 

4.  Divide  586.4  by  375. 


5.  Divide  94.0369  by  81.032. 

6.  Divide  86.2678  by  2.25. 


207.     United  States  Currency. 

If  we  regard  1  dollar  as  the  unit  of  United  States  Cur- 
rency, all  the  lower  denominations, — dimes,  cents,  and  mills, — are 
decimals  of  the  dollar.  Hence,  all  the  operations  upon  United 
States  Money  arc  the  same  as  the  corresponding  operations  on 
decimal  fractions. 


206.  How  do  you  continue  the  division  after  you  have  brought 
down  all  the  figures  of  the  dividend?  When  the  division  does  not 
terminate,  what  sign  do  you  place  after  the  quotient?  What  dous  i' 
show  ? 

207.  What  is  the  unit  of  United  States  Currency?  Wliat  parte  ot 
tiiis  unit  are  dimes?    What  parts  are  cents?    MilU? 


184  CONTRACTIONS. 

CONTRACTIONS    IN    DIVISION. 

208.  Contractions  in  Division  of  Decimals,  like  that  of 
•whole  numbers,  are  short  methods  of  finding  the  quotients, 

CASE     I. 
209.    To  divide  by  10,  100,  &c. 

1.    Divide  479.256  by  10. 

Analysis. — Eemoving  the  decimal  point  one  operation. 

place  to  the  left,   diminishes  the   value  of  the         10)479.256 
decimal  ten  times;  two  places,  100  times,  &c. ;  47.9256 

therefore,   to  divide  by  10,    100,    1000,  &c.,  we 

remove  the  decimal  point  as  many  places  to  the  left  as  there  are 
ciphers  in  the  divisor. 

Rule. — Remove  the  decimal  point  as  many  places  to  the 
left  as  there  are  ciphers  in  the  divisor. 

Examples. 

1.  Divide  3169.274  by  100  ;  by  1000. 

2.  Divide  57135.62  by  1000  ;  by  100  ;  by  10. 

3.  Divide  67.5  by  100  ;  by  1000  ;  by  1000000. 

Note. — If  there  are  not  as  many  figures  at  the  left  of  the  decimal 
point  as  there  are  O's  in  tlie  divisor,  prefix  cipliers  before  writing  the 
decimal  point. 

4.  Divide  4.9  by  100  ;  by  1000  ;  by  10000. 

5.  Divide  .30467  by  10;  by  100  ;  by  1000. 

6.  Divide  .4741  by  100;  by  1000;  by  10000. 

7.  Divide  4.97  by  10  ;  by  100  ;  by  1000. 

CASE    II. 

210.  To  divide  so  that  the  quotient  may  contain  a  given 
number  of  decimals. 

208.  What  are  contractions  in  division  of  decimals? 


DECIMALii.  185 

1.  Divide  754.347385  by  61.34775,  and  find  a  quotient 
which  shall  contain  three  places  of  decimals. 

Rule. 

I.  Note  the  unit  of  the  first  quotient  figure,  and  then  note 
the  number  of  figures  which  the  quotient  must  contain: 

II.  Select,  from  the  left,  as  many  figures  of  the  divisor  as 
you  wish  places  in  the  quotient,  and  multiply  the  figures  so 
selected  by  the  first  quotient  figure,  observing  to  carry  for  the 
figures  cast  off,  as  in  the  contraction  of  multiplication  : 

III.  Use  each  remai?ider  as  a  new  dividend,  and  in  each 
following  division  omit  one  figure  at  the  right  of  the  divisor. 


■  CONTRACTED   METHOD. 

■  61.34775)754.347385(12 
W                        61348 

14086 
12269 

1817 

1227 

690 

652 

38 

37 

1 

296 

COMMON 

61.34775)754.34 
61347 

MKTHOD. 

7385001 
75 

14086 
12269 

988 
550 

1817 
1226 

4385 
9550 

690 
652 

48350 
12975 

38 
36 

353750 
808650 

1 

545100 

Analysis. — In  this  example  the  order  of  the  first  quotient  figure 
is  tens;  hence,  there  are  two  places  of  whole  numbers  in  the 
quotient;  and  as  there  are  three  decimal  places  required,  there 
will  be  five  places  in  all;  hence,  five  figures  of  the  divisor  must 
l)e  used. 

In   the  operation,   by  the   common    method,  the  figures    at  th 
right  of  the  vertical  line,  do  not  aflfect  the  quotient  figures. 

209.  How  do  you  divide  by  10,  100,  &c.?— 210.  Explain  the  manner 
of  dividing,  bo  that  the  quotient  shall  contain  a  given  number  of 
decimal  places. 


186  IlEDUOTION  OF 


Examples. 

1.  Divide  59  by  .*r4571345,  and  let  the  quotient  contain 
four  places  of  decimals. 

2.  Divide  lU93.40n04962  by  495.183269,  and  let  the 
quotient  contain  four  places  of  decimals. 

3.  Divide  98.18743t  by  8.4765618,  and  let  the  quotien  * 
contain  seven  places  of  decimals. 

4.  Divide  4n94.379457  by  14.13495,  and  let  the  quotient 
contain  as  many  decimal  places  as  there  will  be  integers  in  it. 

REDUCTION. 

211.  A  Denominate  Decimal  is  one  in  which  the  unit  of 
the  fraction  is  denominate.  Thus,  .3  of  a  dollar,  .t  of  a  shil- 
ling, .8  of  a  yard,  &c.,  are  denominate  decimals,  in  which  tho 
units  are,  1  dollar,  1  shilling,  1  yard. 

CASE     I. 
212.    To  change  a  common  to  a  decimal  fraction. 

The  value  of  a  fraction  is  the  quotient  of  the  numerator 
divided  by  the  denominator  (Art.  133). 

1.   Reduce  |-  to  a  decimal. 

Analysis. — If  we  place  a  decimal  point  after  the  operation. 

7,   and  then  write  any  number  of  O's  after  it,  the  8)1.000 

value  of  the  numerator  will  not  be   changed.  .875 

If   then,    we    divide    by    the    denominator,    the 
quotient  will  be  the  decimal  number:    Hence, 

Rule. — Annex  decimal  ciphers  to  the  numerator,  and  then 
divide  by  the  denominator,  pointing  off  as  in  division  of 
dedmafs. 

211.  What  is  a  denominate  decimal? 


DECIMALS^  ISt 

Examples. 
Reduce  the  following  common  fractions  to  decimals. 
1.   Reduce  J,  ^,  and  J. 


2.  Reduce  |,  |,  and  ^g-. 

3.  Reduce  |  and  ^^. 

4.  Reduce  y'-j  and  ^*j 

5.  Reduce  J  and  jo'oo- 

6.  Reduce  /j  and  Jf . 

7.  Express  J|-i|  decimally. 

8.  Express  aW?  decimally. 


9.  Reduce  ^-^  and  ^. 

10.  Ex-press  :g\^j  decimally. 

11.  Reduce  -^Vj  and  g^^Q. 

12.  Reduce  |  of  |  of  6. 

13.  Reduce  |  of  \^. 

14.  Reduce  ^^  of  |§. 

15.  Reduce  f  of  f  j. 

16.  Reduce  ^g'  and  yVs- 


n.   What  is  the  decimal  value  of  §•  of  f  multiplied  by  y*^  ? 

18.  What  is  the  value,  in  decimals,  of  ^  of  f  of  J  divided 
by  I  of  J? 

19.  A  man  owns  J  of  a  ship  ;  he  sells  ^  of  his  share  : 
wliat  part  is  that  of  the  whole,  expressed  in  decimals  ? 

20.  Bought  JJ  of  87j\  bushels  of  wheat  for  ^^  of  7  dol- 
lars a  bushel :  how  much  did  it  come  to,  expressed  in  decimals  ? 

21.  If  a  man  receives  |  of  a  dollar  at  one  time,  $7  J  at  an- 
other, and  $8f  at  a  third :  how  much  in  all,  expressed  in 
decimals  ? 

22.  What  mixed  decimal  is  equal  to  the  sum  of  ^  of  18,  j\ 
r  H,  and  7|? 

23.  What  dechnal  is  equal  to  §  of  3|  taken  from  f  of  8|? 

24.  What  decimal  is  equal  to  the  sum  of  Jf,   y,  and  f  ? 

CASE    II. 

■213     To  change  a  decimal  to  the  form  of  a  common  fraction. 
A.NAi.Viiis. — A  ileciiiial  fraction    may  be    changed  to  the  form   ol 
a  common  fraction  by  simply  writing  its  denominator  (Art.  190). 


us, 


REDUCTION    OF 


Examples. 
Express  the  following  decimals  in  common  fractions. 


1.  Reduce  .25  and  .15. 

2.  Reduce  .125  and  .625. 

3.  Reduce  .105  and  .0025. 

4.  Reduce  .8015  and  .6042. 

5.  Reduce  .68375. 


6.  Reduce  .01875. 

7.  Reduce  .22575. 

8.  Reduce  .265625. 

9.  Reduce  .333^ 
10.  Reduce  .5714f. 


CASE    III. 

214.    To  reduce  a  compound  number  to  a  decimal  of  a  given 
denomination. 

1.  Reduce  £1  4s.  9fd.  to  the  decimal  of  a  ig. 


Analysis. — We  first  reduce  3 
farthings  to  the  decimal  of  a 
penny,  by  dividing  by  4.  We 
then  annex  the  quotient  .75d.  to 
the  9  pence.  We  next  divide  by 
12,  giving  .8125,  which  is  the 
decimal  of  a  shilling.  This  we 
annex  to  the  shillings,  and  then 
divide  by  twenty. 


OPERATION. 

fd.  =  .75d. ;  hence, 
9Jd.  =  9.75d. 
12)9.75d. 

.8125s.,  and 
20)4.8125s. 


ie.240625  ;  therefore, 
£1  4s.  9fd.  =  £1.240625 

Rule. — I.  Jf  the  lowest  denomination  contains  a  fraction^ 
reduce  it  to  a  decimal  and  prefix  the  integral  part : 

II.  Then  divide  by  the  units  of  the  ascending  scale,  annex 
the  quotient  to  the  next  higher  dejiomination,  and  proceed  in 
the  same  manner  through  all  the  denominations,  to  the  re- 
quired unit. 

Note. — When  any  denomination,  between  the  lowest  and  the  high- 
est  is  wanting,  the  number  to  be  prefixed  to  the  corresponding  |uo 
tient,  is  0. 

212.  How  do  you  change  a  common  to  a  decimal  fraction? 

213.  How  do  you  change  a  decimal  to  the  form  of  a  common  fraction? 
314.   How  do  you  reduce  a  compound  number  to  a  decimal  of  a 

given  denemination  ? 


DECIMALS.  189 

Examples, 

1.  Reduce  14  drams  to  the  decimal  of  a  lb.  Ayoirdupois. 

2.  Reduce  T8d.  to  the  decimal  of  £. 

3.  Reduce  63  pints  to  the  decimal  of  a  peck. 

4.  Reduce  9  hours  to  the  decimal  of  a  day. 

6.  Reduce  3T5678  feet  to  the  decimal  of  a  mile. 

6.  Reduce  1  oz.  19pwt.  of  silver  to  the  decimal  of  a  pound 

7.  Reduce  3  cwt.  7  lb.  8  oz.  to  the  decimal  of  a  ton. 

8.  Reduce  2.45  shiUings  to  the  decimal  of  a  £. 

9.  Reduce  1.047  roods  to  the  decimal  of  an  acre. 

10.  Reduce  176.9  yards  to  the  decimal  of  a  mile. 

11.  Reduce  2qr.  141b.  to  the  decimal  of  a  cwt. 

12.  Reduce  10  oz.  18pwt.  16gr.  to  the  decimal  of  a  lb. 

13.  Reduce  3qr.  2na.  to  the  decimal  of  a  yard. 

14.  Reduce  1  gal.  to  the  decimal  of  a  hogshead. 

15.  Reduce  17  h.  6  m.  43  sec.  to  the  decimal  of  a  day. 

16.  Reduce  4  cwt.  2Jqr.  to  the  decimal  of  a  ton. 

17.  Reduce  19s.  5d.  2far.  to  the  decimal  of  a  pound. 

18.  Reduce  1  R.  37  P.  to  the  decimal  of  an  acre. 

19.  Reduce  2qr.  3na.  to  the  decimal  of  an  English  ell. 

20.  Reduce  2  yd.  2  ft.  OJ  in.  to  the  decimal  of  a  mile. 

21.  Reduce  15'  22 J"  to  the  decimal  of  a  degree. 

22.  Reduce  1  cwt.  1  qr.  1  lb.  to  the  decimal  of  a  ton. 

23.  Reduce  3  bush.  3pk.  to  the  decimal  of  a  chaldron. 

24.  Reduce  17  yd.  1ft.  6  in.  to  the  decimal  of  a  mile. 

25.  What  decimal  part  of  a  year  is  9 J  months? 

26.  What  decimal  part  of  an  acre  is  IR.  14P.  ? 

27.  What  decimal  part  of  a  chaldron  is  45  pk.? 

28.  What  decimal  part  Of  a  mile  is  72  yards  ? 

29.  What  part  of  a  ream  of  paper  is  9  sheets  ? 

30.  What  part  of  a  rod  in  length  is  4.0125  inches? 

31.  Reduce  10  wk.  2  da.  to  the  decimal  of  a  leap  year. 

32.  Reduce  4  ?    13    13   10  gr.  to  the  decimal  of  a  lb. 

33.  Reduce  3  qt.  1.75  pt.  to  the  decimal  of  a  hhd. 

34.  Reduce  24  sq.  yd.  1.8  sq.  ft.  to  the  decimal  of  an  acre. 


laO  REDUCTION   OF 


CASE     IV. 

215.  To  find  the  value  of  a  decimal  in  integers  of  lower  de- 
nominations. 

1.   What  is  the  value  of  .832296  of  a  iS? 

Analysis. — ^First  multiply  the    decimal  operation. 

by  20,  which  brings  it  to  the  denomination  .832296 

of  shillings,  and  after  cutting  off  from  the  20 

right  as  many  places  for  decimals  as  there  Ifi  fi4'SQ20 

are  in  the  given   number,   we  have  16s.  *         ,„ 

and  the   decimal  .645920  over.      This  is  

reduced  to  pence  by  multiplying  by  12,  7.751040 

and    then    to    farthings    by    multiplying  4 

^y  ^-  3^004160 

Ans.  16s.  7d.  3far. 
Rule 

I.  Multiply  the  decimal  by  the  units  of  the  descending 
scale,  and  point  off  as  in  the  multiplication  of  decimals : 

II.  Multiply  the  decimal  part  of  the  product  as  before,  and 
continue  the  operations  to  the  lowest  denomination.  The  in- 
tegers cut  off  at  the  left,  form  the  answer. 

Examples. 

1.  What  is  the  value  of  .6725  of  a  hundredweight? 

2.  What  is  the  value  of  .61  of  a  pipe  of  wine? 

3.  What  is  the  value  of  .83229  of  a  iS? 

4.  Requu-ed  the  value  of  .0625  of  a  barrel  of  beer. 
6.  Required  the  value  of  .42857  of  a  month. 

6.  Required  the  value  of    .05  of  an  acre. 

7.  Required  the  value  of  .3375  of  a  ton. 

8.  Required  the  value  of  .875  of  a  pipe  of  wine. 

9.  What  is  the  value  of  .375  of  a  hogshead  of  beer  ? 

al5.  How  do  you  find  the  value  of  a  decimal  in  integers  of  lower 
denominations  ? 


DKCIMALS.  191 

10.  What  is  the  value  of  .911111  of  a  pound  troy? 

11.  What  is  the  value  of  .G75  of  an  English  ell? 

12.  What  is  the  value  of  .001136  of  a  mile  in  length? 

13.  What  is  the  value  of  .000242  of  a  square  mile? 

14.  Required  the  value  of  .4629  degrees. 

15.  Required  the  value  of  .8t5  of  a  yard. 

16.  Required  the  value  of  .3489  of  a  pound,  apothecaries. 

17.  Required  the  value  of  .759  of  an  acre. 

18.  Required  the  value  of  .01875  of  a  ream  of  paper. 

19.  Required  the  value  of  .0055  of  a  ton. 

20.  Required  the  value  of  .625  of  a  shilling. 

21.  Required  the  value  of  .3375  of  an  acre.. 

22.  Required  the  value  of  .785  of  a  year  of  365^  days. 

REPEATING    DECIMALS. 

216.  In  changing  a  common  to  a  dechnal  fraction,  there  are 
two  general  cases  : 

1st.  When  the  division  terminates  ;  and 

2d.  When  it  does  not  terminate. 

In  the  first  case,  the  quotient  will  contain  a  limited  number 
of  decimal  places,  and  the  exact  value  of  the  common  fraction 
will  be  expressed  decimally. 

In  the  second  case,  the  quotient  will  contain  an  infinite 
number  of  decimal  places,  and  the  exact  value  of  the  common 
fraction  cannot  be  expressed  decimally. 

CASE     I. 

217.     When  the  division  terminates. 

When  a  common  fraction  is  reduced  to  its  lowest  terms 
(wliich  we  suppose  to  be  done  in  all  cases  that  follow),  there 
will  be   no  factor  common   to   its   numerator  and  denominator, 

216.  How  many  cases  are  there  in  changing  a  common  to  a  deci 
mal  fraction?  What  are  they?  What  distinguishes  one  of  these 
cases  from  tho  other? 


192 


REPEATING 


1.  Reduce  -J  J  to  its  equivalent  decimal. 

Analysis. — Annexing  one  0  to  the  numer- 
ator multiplies  it  by  10,  or  by  2  and  5 ; 
hence,  2  and  5  become  prime  factors  of 
the  numerator  every  time  that  a  0  is  an- 
nexed. But  if  the  division  is  exact,  these 
prime  factors,  and  none  others^  must  also  be 
found  in  the  denominator. 

2.  Reduce  /g  to  its  equivalent  decimal. 

Analysis.— 36=18  x2  =  9x2x2  = 
8x3  X  2  X  2;in  which  we  see  that  the 
denominator  contains  other  factors  than  2 
and  5;  hence,  the  fraction  cannot  he  ex- 
actly expressed  decimally. 


OrERATION. 

50)11.00(.34 
150 
2  00 
2  00 


OPERATION. 

36)5.0(.1388  + 
36 
140 
108 


320 


320 

288 


Rule. — I.  Reduce  the  fraction  to 
its  lowest  terms,  then  decompose  the  de- 
nominator into  its  prime  factoids;  and 
if  there  are  no  factors  other  than  2 
and  5,  the  exact  division  can  be  made: 

II.  If  there  are  other  prime  factors^  the  exact  division  can- 
not he  made. 

N«TE. — Every  0  annexed  to  the  numerator,  introduces  the  two 
factors  2  and  5;  and  these  factOTS  must  be  introduced  until  we  have 
as  many  of  each  as  there  are  in  the  denominator  after  it  shall  have 
been  decomposed  into  its  prime  factors  2  and  5.  But  the  quotient 
will  contain  as  many  decimal  places  as  there  are  decimal  O's  in  the 
dividend.    Hence, 

The  number  of  decimal  2'>lcices  in  the  quotient  icill  be  equal 
to  the  greatest  number  of  factors^  2  or  5,  ^V^  the  divisor. 

3.  Can  ^^  be  exactly  expressed  decimally  ?  How 
many  places  ? 

Analysis. — 25  =5x5;  hence,  the  fraction  can  be  ex- 
actly expressed  decimally,  and  by  two  decimals,  because 
5  is  taken  twice  as  a  factor  in  the  divisor. 


operation. 

25)7.0(.28 
50 

200 
200 


DECIMALS.  193 

Examples. 
Find  the  decimals  and  number  of  places  in  the  following : 


1.  Express  yf^  decimally. 

2.  Express  -^^q  decimally. 

3.  Express  ^^  decunally. 

4.  Expicss  ji^  decimally. 


5.  Express  -^  decimally. 

6.  Express  ^lt^  decunally. 
T.  Express  -^  decimally. 
8.  Express  ^f^^  decunally. 


CASE     II. 
218.    When  the  division  does  not  terminate. 

1.  Let  it  be  required  to  reduce  J  to  its  equivalent  decimal. 

Analysis. — ^By  annexing  decimal  ciphers  to  the  operation. 
numerator  1,  and  making  the  division,  we  find  the  3)1.0000 

equivalent  decimal  to  be  .3333  +  ,  «&c.,  giving  3's  ^333  + 

as  far  we  choose  to  continue  the  division. 

The  further  the  division  is  continued^  the  nearer  the  value 
of  the  decimal  will  approach  to  \,  the  exact  value  of  the  com- 
mon fraction.  We  express  this  approach  to  equality  of  value, 
by  saymg,  that  if  the  division  be  continued  without  limits  that 
is,  to  infinity,  the  value  of  the  decimal  will  then  become  equal 
to  that  of  the  common  fraction  j  thus, 

.3333 .. .  ,  continued  to  infinity  =  J ; 
for,  each  succeeding  3  brings  the  value  nearer  to  \. 

Also,  .9999 . . .  ,  continued  to  infinity  =  1  ; 

for,  each  succeeding  9  brings  the  value  nearer  to  1, 

2.  Find  the  decimal  corresponding  to  the  common  fraction  §. 

Analysis. — Annexing     decimal     ciphers     and         operation.  • 

dividing,   wo    find  the  decimal  to  ho  .2222+,.  9)2.0000 

in  which  we  see  that  the  figure  2  is  continually  .2222 

repeated. 

9 


L 


194-  K'iPEATlNO 

Examples. 

1.  Express  the  fraction  f  decimally. 

2.  Change  1*3-  into  a  decimal  fraction. 

3.  Reduce  j\  to  a  decimal  fraction. 

4.  Reduce  -j^  to  a  decimal  fraction. 

219.   Definitions. 

1.  A  Repeating  Decimal  is  a  decimal  in  which  a  sin(,le 
figure,  or  a  set  of  figures,  is  constantly  repeated. 

2.  A  Repetend  is  a  single  figure,  or  a  set  of  figures,  which 
is  constantly  repeated. 

3.  A  Single  Repetend  is  one  in  which  only  a  single  figure 
Is  repeated  ;  as 

j  =  .2222+,   or  f  =  .3333+. 

Such  rcpetends  are  expressed  by  simply  putting  a  mark  over 
the  first  figure  ;  thus, 

.2222+,  is  denoted  by  .^2,  and  .3333+  by  .^3. 

4.  A  Compound  Repetend  has  a  set  of  figures  repeated  ; 
thus, 

If  =  .5t  5t+,  and  jj|3  ^  5^33  5123  + 

are  compound  repetends,  and  are  distinguished  by  marking  the 
first  and  last  figures  of  the  set.  Thus,  51  57+  is  written 
.^57',  and  .5723  5723  +  is  written  .^5723'. 

5.  A  Pure  Repetend  is  one  which  begins  with  the  first 
decimal  figure  ;   as, 

.^3,         .^5,         .^473',         &c. 

217.  How  do  you  determine  wlien  a  common  fraction  can  be  ex 
actly  expressed  decimally?  How  many  decimal  places  will  there  bo 
in  the  quotient ?— 218.  Can  one-third  be  exactly  expressed  decimally? 
Wliat  is  the  form  of  tlie  quotient?  To  what  does  the  value  of  this 
quotient  approach?  Wlien  does  it  become  equal  to  one-third ?— 219. 
1.  What  is  a  repeating  decimal?  2.  What  is  a  repetend?  3.  What 
is  a  single  re^xitend?    4.  What  is  a  compniud  repetend? 


DECIMALS.  19o 

6.  A  Mixed  Repetexd  is  one  which  has  significant  figures 
or  ciplicrs  between  the  decimal  point  and  the  repetend ;  or 
which  has  whole  numbers  on  the  left  hand  of  the  decimal 
point  ;  such  figures  are  called  finite  figures.    Thus, 

.0^733',     .4^73;     .3^573',     6/5, 
are  all  mixed  repetends  ;  .0,  .4,  .3,  and  6,  are  the  finite  figures 

7.  Similar  Repetends  are  such  as   begin   at   equal  distance 
from  the  decimal  points  ;  as  .3^54',  2.7'534'. 

8.  Dissimilar  Repetends  are  such  as  begin  at  different  dis- 
tances from  the  decimal  points  ;  as  .'^253',  .47'52'. 

9.  Conterminous  Repetends  are  such  as  end  at  equal  dis- 
tances from  the  decimal  points ;  as  .1^25',  /354'. 

10.  Similar  and  Conterminous  Repetends  are  such  as  begin 
and  end  at  the  same  distances  from  the  decimal  point ;  thus, 
53.2V753',  4.6^325',  and  .4^632',  are  similar  and  conterminous. 

REDUCTION    OF  REPETENDS  TO  COMMON  FRACTIONb. 

CASE     I. 

220.  To  reduce  a  pure  repetend  to  its  equivalent  common 
fraction. 

Analysis. — This  proposition  is  to  be  analyzed  hy  examining 
the  law   of  forming  the  repetends. 

let.    ^=.llll+«&c.  =  .n;  and  f  =  4444+ &c.  =  .H: 

2d.  ^V  =  . 010101+  &c.  =  .^01';     and  f  J  =  .2727+  &c.  =  .•»27': 
3d.  ^Jy  =.001001  +  &c.=.^001' ;  and  f ff  =  .324324+  &c.  =  .^324'; 

&c.,  &c.,  &c.,  &c. 

The  above  law  for  the  formation  of  repetends  does  not  depend 

on  the  multipliers  4,  27,  and  324,  but  would  be  the  same  for  any 

other  figures. 

Rule. — Divide  the  number  denoting  the  repetend  by  as 
many  9's  as  there  are  figures,  and  reduce  the  fraction  to 
its  loiuest  terms. 


196  KEPEATING 

Examples. 

1.  What  is  the  equivalent  common  fraction  of  the  repetecd 
0.^3? 

We  have,       f  =  J  =  0.33333+  =  ^3. 

2.  What  is  the  equivalent  common  fraction  of  the  repeten 
.n62'  ? 

We  have,  Jf  J  =  j\\.  Ans. 

3.  What  are  the  simplest  equivalent  common  fractions  of  the 
repetends  /6,  /162',  0;t69230',  /945',  and  /09'? 

4.  What  are  the  least  equivalent  common  fractions  of  the 
repetends-  /594405',  .^36',  and  .^142857'  ? 

CASE     II. 

221.  To  reduce  a  mixed  repetend  to  its  equivalent  common 
fraction. 

Analysis. — A  mixed  repetend  is  composed  of  the  finite  figures 
which  precede,  and  of  the  repetend  itself;  hence,  its  value  must 
be   equal  to  such  finite  figures  plus  the  repetend. 

When  the  repetend  begins  at  the  decimal  point,  the  unit  of 
the  first  figure  is  .1.  But  if  the  repetend  begins  at  any  place  at 
the  right  of  the  decimal  point,  the  unit  value  of  the  first  figure 
will  be  diminished  ten  times  for  each  place  at  the  right,  and 
hence,   O's  must  be   annexed  to  the   9's  which  form  the   divisor. 

Rule. — To  the  finite  figures,  add  the  repetend  divided  by 
as  many  9's  as  it  contains  places  of  figures,  with  as  many 
O's  annexed  to  them  as  there  are  places  of  decimal  figures 
vreceding  the  repetend ;  the  sum  reduced  to  its  simplest  form 
will  be  the  equivalent  fraction  sought. 

219.  5.  What  is  a  pure  repetend  ?  6.  What  is  a  mixed  repetend 
7.  What  are  similar  repetends?  8.  Wliat  are  dissimilar  repetends? 
9.  What  are  conterminous  repetends?  10.  What  are  similar  and 
conterminous  repetends  ? — 220.  How  do  you  reduce  a  pure  repetend  to 
an  equivalent  common  fraction? — 321.  How  do  you  find  the  value  of 
a  mixed  repetend? 


DECIMALS.  197 

Examples. 

1.  Rcqiured  the  least  equivalent  common  fraction  of  the 
mixed  repetend,  2.4^18'. 

Now, 
2.4^18'  =  2  +  ^V  +  '18'  =  2  +  A  +  9'd%  =  2Ii.  Ans. 

2.  Required  tM)  least  equivalent  common  fraction  of  the 
mixed  rcpetcud  .5'925'. 

We  have,      .5^925'  =  tV  +  qWo  =  if  ^l"^'- 

3.  What  is  the  least  equivalent  common  fraction  of  the 
repetend  .008^497133'? 

We  have,    .008^497133'  =  ^^'oo  +  99W9W00  =  ms- 

4.  Required  the  least  equivalent  common  fractions  of  the 
mixed  repetends  .13^8,  7.5^43',  .04^354',  37.5^4,  .6^75',  and 
.7^54347'. 

5.  Required  the  least  equivalent  common  fractions  of  the 
mixed  repetends  0.7^5,  0.4^38',  .09^3,  4.7^543',  .009^87',  and  .4^5. 

CASE      III. 

222.  To  find  the  finite  figures  and  the  repetends  correspond- 
ing to  any  common  fraction. 

1.  Find  the  finite  figures  and  the  repetend  corresponding  to 
the  fraction  ^Iq. 

Analysis. — 1st.  Reduce  the 
fraction  to  its  lowest  terms,  and 
tlien  find  all  the  factors  2  and  5 
of  the  denominator. 

2d.  Add  decimal  ciphers  to 
tlie  numerator  and  make  the 
division. 

8d.  The  number  of  Jinite  decimals  preceding  the  first  figure 
of  tlie  repetend  will  he  equal  to  the  greatest  number  of  factors 
2  or  5:  in  this  example  it  is  3. 


OPERATION, 

6 

560  ~ 

3 

'   280 

3 

3 

280 

~2x2 

X2x5x7 

280)3.000+  (. 

010^714285' 

lOS  REPEATING 

4th.  When  a  remainder  is  found  which  is  the  same  as  a  previous 
dividend,  the  second  repetend  begins. 

5th.  The  number  of  figures  in  any  repetend  will  never  exceed 
the  number,  less  1,  of  the  units  in  that  factor  of  the  denominator 
which  does  not  contain  2  or  5.  In  the  example,  that  number  is  7, 
and  the  number  of  figures  of  the  repetend.  is  6. 

Rule. — Divide  the  numerator   of  the  common  fraction,  r 
duced   to  its  lowest  terms,  by  the   denominator,  and  poini  o// 
in  Ike  quotient   the  finite  decimals,  if  any,  and  the  rep:  te-iui. 


Examples. 

1.  Find  whether  the  decimal,  equivalent  to  the  common 
fraction  29 f  oTj  ^^  finite  or  repeating :  required  the  finite 
figures,  if   any,  and  the  repetend. 

Analysis. — We  first  reduce  the 
fraction  to  its  lowest  terms,  giving 
gfffif*  ^^^©  then  search  for  the 
factors  2  and  5  in  the  denominator, 
and  find  that  2  is  a  factor  3  times ; 
hence,  we  know  that  there  are 
three  finite  decimals  preceding  the  q*7A8\qq  qq  ■  (  008^49*riSS' 
repetend.      We    next    divide    the 

numerator  83  by  the  denominator  9768,  and  note  that  the  repetend 
begins  at  the  fourth  place.  After  the  ninth  division,  we  find  the 
remainder  83;  at  this  point  the  figures  of  the  quotient  begin  to 
repeat ;  hence,  the  repetend  has  6  places. 

2.  Fmd  the  finite  decimals,  if  any,  and  the  repetend,  if  any, 
of  the  fraction  yV&* 

3.  Find  the  finite  decimals,  if  any,  and  the  repetend,  if  any, 
of  the  fraction  xieo- 

4.  Find  the  finite  decimals,  if  any,  and  the  repetend,  if  any, 
cf  the  fractions  ^ft,  tVj,  tVs- 


OPERATION. 

249             83 
29304         9768 

83 

83 

9768 

2x 

;2x2xl221 

DECIMALS.  100 

223.    Properties  of  the  Repetends. 

There  are  some  properties  of  repetends  which  it  is  important 
to  remark. 

1.  Any  finite  decimal  may  be  considered  as  a  repeatmg 
decimal  by  making  ciphers  recur  ;  thus, 

.35  =  .35^0  =  .35^00'  =.35^000'  =  .35^0000',  &c. 

2.  Any  repeating  decimal,  whatever  its  number  of  figures, 
may  be  changed  to  one  having  twice  or  thrice  that  number  of 
figures,  or  any  multiple  of  that  number. 

Thus,  a  repetend  2.3^57'  having  two  figures,  may  be  changed 
to  one  having  4,  6,  8,  or  10  places  of  figures.    For, 

2.3^5r  =  2.3^575r  =2.3^575757'  =  2.3^57575757',  &c. ; 
so,  the  repetend  4.16'316'  may  be  written 

4.16^316'  =  4.16^316316'  =  4.16^316316316',   &c. ; 
and   the   same   ma^  be   shown   of   any  other.     Hence,  two   or 
more   repetends,  having  a  different  number   of  places   in   each, 
may  be  reduced  to  repetends  having  the  same  number  of  places, 
in  the  following  manner : 

Find  the  least  common  multiple  of  the  number  of  places  in 
each  repetend,  and  reduce  each  repetend  to  such  number  of 
places, 

3.  Any  repeating  decimal  may  be  transformed  into  another 
having  finite  decimals  and  a  repetend  of  the  same  number  of 
figures  as  the  first.     Thus, 

.^57'  =  .5^5'  =  .57^57'  =  .575^75'  =  .5757^57' ;  and 
3.4^785'  =  3.47^857'  =  3.478^578'  =  3.4785^785' ; 

and  hence,  any  two  repetends  may  be  made  similar. 

222.  How  do  you  find  the  finite  figures  and  the  repetend  corro- 
pponding  to  any  common  fraction? — 223.  1.  How  may  a  finite  decimal 
be  made  a  repeating  decimal?  S.  Wlien  a  rejpetend  has  a  given 
number  of  places,  to  what  other  form  may  it  be  reduced?  How? 
3.  Into  what  form  may  any  repeating  decimal  be  transformed? 


L 


200  REPEATING 

These  properties  may  be  proved  by  changing  the  repetends 
into  their  equivalent  common  fractions. 

4.  Having  made  two  or  more  repetends  similar  by  the  last 
article,  they  may  be  rendered  conterminous  by  the  previous 
one  ;  thus,  two  or  more  repetends  may  always  he  made  similar 
nd  conterminous. 

5  If  two  or  more  repeatmg  decunals,  having  several  repe- 
tends of  equal  places,  be  added  together,  their  suu>  will  have 
a  repetend  of  the  same  number  of  places  ;  for,  every  two  sets 
of  repetends  will  give  the  same  sum. 

6.  If  any  repeatmg  decimal  be  multiphed  by  any  number, 
the  product  will  be  a  repeating  decimal  having  the  same 
number  of  places  in  the  repetend  ;  for,  each  repetend  will  be 
taken  the  same  number  of  times,  and  consequently  must  pro- 
duce the  same  product. 

Examples. 

1.  Reduce  .13^8,  Y.5'43'  .04^354',  to  repetends  havmg  the 
same  number  of  places. 

Since  the  number  of  places  are  now  1,  2,  and  3,  the  least 
common  multiple  is  6,  and  hence  each  new  repetend  will  con- 
tain 6  places  ;  that  is, 

.13^8  =  .13^888888' ;  T.5^43'  =  7.5^434343' ;  and 
.04^354'  =  .04^354354'. 

2.  Reduce  2.448',  .6^925',  .008^497133',  to  repetends  having 
the  same  number  of  places. 

3.  Reduce  the  repeating  decunals  165.464',  .^04',  .03^7  to 
such  as  are  similar  and  conterminous. 

4.  Reduce  the  repeating  decunals  .5'3,  .4^75',  and  1.^757', 
o  such  as  are  similar  and  conterminous. 

223.  4.  To  wliat  form  may  two  or  more  repetends  be  reduced? 


DECIMALS.  201 

ADDITION. 
224.    To    add    repeating   decimals. 

I,  Make  the  repetends,  in  each  number  to  be  added,  similar 
and  conterminous  : 

II.  Write  the  places  of  the  same  unit  value  in   the  same 
'olumn,  and  so  many  figures  of  the  second  repetend  in  each 

as  shall  indicate  xdlh  ceirtainty,  how  many  are  to  be  carried 
from  one  repetend  to  the  other :  then  add  as  in  whole  numbers. 

Note. — If  all  the  figures  of  a  repetend  are  9's,  omit  them  and  add 
1  to  the  figure  next  at  the  left. 

Examples. 
1.   Add  .12^5,  4.U63',  1.^7143',  and  2.^54',  together. 

DLSSIMILAR.        SIMILAR.  SIMILAR   AND   COXTERMINOUS. 

.12^5  =    .12'5         =    .12^555555555555'  -  -  -  5555 

4.n63' =  4.10^316'    =  4.1G^31631G316316'  -  -  -  31C3 

1.^7143'  =  1.7r4371'  =  1.7^437143714371'  -  -  -  4371 

2.^54'  =  2.54^54'      =  2.54^545454545454'  -  -  -  5454 


The  true  sum  =  8.54^854470131697'  1  to  carry. 

2.  Add  67.3^45',  9.^G51',  .^25',  17.4^,  .\5,  together. 

3.  Add  .^475',  3.75^43',  64.^5',  .^57',  .r788',  together. 

4.  Add  .^5,  4.3^7,  49.4^57',  .4^954',  .^7345,'  together. 

5.  Add  .^175',  42.^57',  .3^753',  .4^954',  3.7^54',  together. 

6.  Add  165,   .464',    147.^04',    4.^95',  94.3^    4.^12345'. 

SUBTRACTION. 
225.     To    subtract   one   repeating  decimal  from  another. 

I.  3Iake  the  repetends  similar  and  conterminous : 

II.  Subtract  as  in  finite  decimals,  observing  that  when  the 
repetend  of  the  lower  line  is  the  larger,  1  must  be  carried  to 
the  first  right-hand  figure. 

224.  How  do  you  add  repeating  decimals? — 225.  How  do  you  sub- 
tract repeating  decimals. 

9* 


202 


REPEATING 


Examples 
1.   From  11.4^75'  take  3.45^35'. 

DISSIMILAR.  SIMILAR.  SIMILAR  AND  CONTERMINOUS. 

11.4^75'  =  11.47^5r  =  11.4r575757'     -     - 
3.45^35'  =  3.45^735'  =    3.45^35735'     -     - 


575 

735 


The  true  difference  =    8.0r840021'  1  to  carry. 


2.  From  47.5^3  take  i;757'. 

3.  From  17.^573'  take  14.5^7. 

4.  From  17.4^3  take  12.34^3. 

5.  From  1.12^754'  take  .4^384'. 


6.  From  4.75  take  .37^5. 

7.  From  4.794  take  .1^744'. 

8.  From  1.45^7  take  .3654. 

9.  From  1.4^937'  take  .1475. 


MULTIPLICATION. 
226.      To  multiply  one  repeating  decimal  by  another. 

Change  Ihe  repeating  decimaU  into  their  erjaivalent  com 
moii  fraetiom^,  then  multiply  them  together,  and  reduce  tht 
product  to  its  equivalent  repealing  decimal. 


Examples. 
1.   Multiply  4.25^3  by  .257. 

OPKUATION. 

4  '>ry3    —   4    4-   -?5.    _!_   __3  _    _   4      I      22  5      I  3_    __ 

*•-•'  "^   —   ^      '      100      "^aOO    —    ^    ^    900    ^900    — 


3_     __    22  8 
»00 


3_8  2_8 
900 


L9J  4 

4  50 


957 
2  2.5 


Also. 


^25     -^ 


257  —  -^^"^    ♦   hencf 

.-/.J  i    —    100  0    5     "^"t.1,, 
1000 


MiUl  =  1.09310^0  ; 


and  since  225000  =  5x5x5x5x5x2x2x2x9, 
there  will  be  five  places  of  finite  decimals,  and  one  figure  in 
the  repeteiid. 

Note. — Much  labor  will  be  saved  in  this  and  the  next  rule  by  keep- 
ing every  fraction  in  its  lowest  terms ;  and  when  two  fractions   are  to 
be  multiplied  together,  cancel  all  the  factors  common  to   both  term 
before  making  the  multiplication. 


2.  Multiply  .375^4  by  14.75. 

3.  Multiply  .4^253'  by  2.57. 

4.  Multiply  .437  by  3.7^5. 

5.  Multiply  4.573  by  .3^75'. 


6.  Multiply  3.^5^6  by  .42^5. 

7.  Multiply  1.H56'  by  4.2^3. 

8.  Multiply  45.r3   by  .^245'. 

9.  Multiply  .4705^3  by  1.7^35'. 


DECIMALS.  203 

DIVISION. 

227.    To  divide   one  repeating  decimal   by   another. 

Change  the  decimals  into  their  equivalent  common  fractionb, 
and  find  the  quotient  of  these  fractions,  Tlien  change  the 
quotient  into  its  equivalent  decimal. 

Examples. 
1.   Divide  5G.^6  by  13t. 

OPERATION. 

56.^0  =  56  +  f  =  ^^  =  ip. 
Then,    i^  ^  137  =  ip  x  yir  =  iiT  =  .'41362530'. 


2.  Divide  24.3^18'  by  1.192. 

3.  Divide  8.59G8  by  .2^45'. 

4.  Divide  2.295  by  .^29r. 

5.  Divide  47.345  by  1.^6'. 


6.  Divide  13.5469533' by  4.^29t' 
1.  Divide  .^45' by  .418881'. 

8.  Divide  .^475'  by  .3^753'. 

9.  Divide  3.^153'  by  .^24'. 


CONTINUED    FRACTIONS. 

228.  A  Continued  Fraction  has  1  for  its  numerator,  and 
for  its  denominator  a  whole  number  plus  a  fraction,  which  also 
has  a  numerator  of  1,  and  for  a  denominator,  a  whole  number 
plus  a  similar  fraction,  and  so  on. 

.1.    If  we  take   any   irreducible   fraction,    as   ^J,  and  'divide 
both  terms  by  the  numerator,  it  will  take  the  form, 

'       ^  =  2?  =  1    ,    14    ^^y  making  the  division. 

If,  now,  we  divide  both  terms  of  ||^  by  14,  we  have, 

14__1 

15-1+tV 

220.  How  do  you  multiply  repeating  decimals? — 237.  How  do  you 
divide  repeating  decimals ?— 228.  What  arc  continued  fractions?  What 
hi  the  rule  for  finding  tha  approximate  value? 


204:  CONTINUED 

If.  now,  we  replace  ^f  3y  its  value,  ,- ,  we  shall  liave 

1   +  14 

15       1 


29       1  +  1 


1+A; 

hence,  this  is  a  '.ontinued  fraction, 

2.   Reduce  \^  to  the  form  of  a  continued  fraction. 


15 
19" 

1 
"1  + 

15 
19 

4        1 

15  -  3  +  V 

1 
~1+1 

3  +  1 
1  + 

3 
4 

I. 

l  +  i 

^p^        1 

hence. 


Analysis. — Let  us  analyze  this  example.  If  we  neglect  what 
comes  after  1,  the  first  term  of  the  first  denominator,  we  shall 
have,  T  =  1?  which  is  called  the  first  approximating  fraction.  If  we 
neglect  what  comes  after  3,  the  first  term  of  the  second  denomi- 
nator, we  shall  have, 

1      _    3 
1+1-4 
the  second  approximating  fraction. 

If  we  neglect  what  comes  after  1,  the   first  term  of  the   third 
denominator,  we  shall  have, 
1 


1+1  =-5 

3  +  1 

the  third  approximating  fraction;    and  so    on,  for  fractions  which 

follow. 

If  we  stop  at  the  first  approximating  fraction,  the  denominator  1 
will  he  less  than  the  true  denominator;  for,  the  true  denominator 
is  1  plus  a  fraction;  hence,  the  value  of  the  first  approximating 
fraction  will  he  too  great;  that  is,  it  will  exceed  the  value  of  the 
given  fraction. 

If  we  stop  at  the  second,  the  denominator  3  will  he  less  than  the 
true  denominator;  hence,  ^  will  he  greater  than  the  number  to  ho 
ad'lcd    to    1  ;    therefore,  1  +  /f   is    too  large,  and    1  -f-  1  +  5,  wliioli 


FRACTIONS.  205 

is  J,  is  too  small :  that  is,  it  is  less  than  the  value  of  the  given  frac- 
tion. Thus,  every  odd  approximating  fraction  gives  a  value  too 
large,  and  every  even  one,  gives  a  value  too  small. 

Rule. —  Write  the  given  fraction  in  the  form  of  a  continued 
fraction,  using  several  terms  when  a  near  approximation  is 
desired ;  then  take  a  mean  betiveen  the  last  and  the  preceding 
approximating  fractions. 

Examples. 
1    Reduce  4^1  to  the  form  of  a  continued  fraction. 

437 


2  +  1 


l  +  l 


1  +  1 


3  +  tV. 

2.  Place  f  J   under   the   form   of   a   continued   fraction,  and 
find  the  value  of  each  of  the  approximating  fractions. 

3.  Place   U   under   the   form   of  a   continued   fraction,  and 
find  the  value  of  each  of  the  approximating  fractions. 

4.  Place  \\  under  the  form  of  a  continued  fraction,  and  find 
the  value  of  each  approximating  fraction. 

5.  Place  f  J  under  the  form  of  a  continued  fraction,  and  find 
the  value  of  each  approximating  fraction. 

G.   Place  -J^  under  the  form  of  a  continued  fraction,  and  find 
the  value  of  each  approximating  fraction. 

7.  The  solar  year  contains  365  da.  5  hr.  48  m.  48  sec.    Find 
what  fractional   part  of  a  day  the  excess  of  the  solar  year  is 
bove  the   common  year,  when  the  operation  is  carried  to  the 
fifth   approximating  fraction. 


206  RATIO   AND   PKOPORTION. 


RATIO    AND    PROPORTION. 

229.  Two  numbers,  of  the  same  kind,  may  be  compared  in 
two  ways  : 

1st.  By  considering  how  much  one  is  greater  or  less  thai 
the  other,  which  is  shown  by  their  difference  ;  and, 

2d.  By  considering  how  mafiy  times  one  number  is  greater 
or  less  than  another,  which  is  shown  by  their  quotient. 

In  comparing  two  numbers,  by  means  of  their  difference,  the 
less  is  always  taken  from  the  greater. 

In  comparing  two  numbers  by  their  quotient,  one  is  regarded 
AS  a  standard  which  measures  the  other  ;  hence,  to  measure 
a   number,  is  to  find  how  many  times  it  contains  the  standard. 

230.  A  Ratio  is  the  quotient  arising  from  dividing  one  num- 
ber by  another. 

The  Terms  of  a  ratio  are  the  divisor  and  dividend  ;  hence, 
every  ratio  has  two  terms. 

The  Divisor  is  called  the  Antecedent  ;  and  the  Dividend  is 
called  the  Consequent. 

The  antecedent  and  consequent,  taken  together,  are  called 
a  Couplet. 

231.  The  ratio  of  one  number  to  another  is  expressed  in  two 
ways  : 

1st.  By  a  colon  ;  thus,  4  :  16  ;  and  is  read,  4  is  to  16  ;  or, 
16  divided  by  4. 

2d.  In  a  fractional  form,  as  Y  >  <^^>  1^  divided  by  4. 

229.  In  how  many  ways  may  two  numbers  of  the  same  kind  be 
compared  with  each  other?  If  you  compare  by  their  difference,  wliat 
do  you  do  ?  If  you  compare  by  the  quotient,  how  do  you  regard  one 
of  the  numbers?    How  do  you  measure  a  number? 

230.  What  is  a  ratio?  What  are  its  terms?  How  many  terms  lias 
every  ratio?  What  is  the  divisor  called ?  What  the  dividend ? — 231.  In 
how  many  ways  is  the  ratio  expressed?  What  are  they?  How  is  it 
read  ? 


RATIO    AND    niOFORTION.  207 

Since  every  ratio  may  be  expressed  under  the  form  of  a 
fraction,  and  since  tlie  numerator  and  denominator  may  be  mul- 
tiplied or  divided  by  the  same  number,  without  altering  the 
value  (Arts.  143  and  144),  it  follows  that. 

If  both  terms  of  a  ratio  be  multiplied  or  divided  by  the 
mme  number,  the  ratio  will  not  be  changed. 

232.  A  Simple  Ratio  is  when  both  terms  are  simple  num 
bers  ;  thus, 

t  :  12,  is  a  simple  ratio. 

233.  A  Compound  Ratio  is  one  which  arises  from  the  mul- 
tiplication of  two  simple  ratios  :  thus,  in  the  simple  ratios 

5     :     10,    and    3     :     12, 
if  we  multiply  the  corresponding  terms  together,  we  have 

5      X     3     :     10     X      12, 
whicli  is  compounded  of  the  ratios  of  5  to  10,  and  of  3  to  12, 

234.  The  Elements  of  a  term  are  its  factors  :  thus,  5  and  3 
are  the  elements  of  the  first  term,  and  10  and  12  of  the  second. 

These  elements  are  generally  WTitten  in  a  column,  thus, 

!!  [-      :     1 .)  f  ;  ^"^^  read,  5  multiplied  by  3,  to  10  multiplied  by  12. 

Note. — A  compound  ratio  may  be  reduced  to  a  simple  ratio,  by 
multiplying  the  elements;  thus  the  last  ratio  is  that  of  15  to  130. 

235.     To  find  the  ratio  of  one  number  to  another. 

AN'hcn  the  antecedent  is  less  than  the  consequent,  the  ratio 
>li(jws  how  many  times  the  consequent  is  greater  than  the 
•  ntecedent. 

When  the  antecedent  is  greater  than  the  consequent,  the 
ratio  shows  what  part  the  consequent  Ls  of  the  antecedent. 
The  phrase,  "what  part,"  implies  the  quotient  of  a  less  num- 
ber divided  by  a  greater. 

233.  What  is  a  simple  ratio?— 333.  What  is  a  compound  ratio?— 
23 i.  What  are  the  elements  of  a  term? 


208 


KATIO  AND   PKOPORTION". 


Examples. 
1.   What  is  the  ratio  of  9  tons  to  15  tons  ? 

Analysis. — In  this  example  the   antecedent  is  9  tons,  and  the 
consequent  is  15  tons ;  the  ratio  is  therefore  expressed  by  the  frac- 


2.  What 

3.  What 

4.  What 

5.  What 

6.  What 
t.  What 

8.  What 

9.  What 

10.  What 

11.  What 

12.  What 


is  the  ratio  of  6  inches  to  24  inches  ? 
is  the  ratio  of  1  feet  to  35  feet? 
is  the  ratio  of  fifteen  dollars  to  6  dollars  ? 
is  the  compound  ratio  of  5  :  6  and  4  :  10  ? 
is  the  compound  ratio  of  6  :  9  and  3:4? 
is  the  compound  ratio  of  4  :  5,  9  :  8,  and  3:5? 


part  of  6  is  4? 
part  of  10  is  5? 
part  of  34  is  It  ? 
part  of  450  is  300? 
part  of  96  is  16? 


13.  8  is  what  part  of  12  ? 

14.  16  is  what  part  of  48? 

15.  18  is  what  part  of  90  ? 

16.  15  is  what  part  of  165? 
It.  9  is  what  part  of  11? 


236.    To  find  the  antecedent  or  consequent,  when  the  ratio  and 
one  of  the  terms  are  given. 

1.    The  ratio  of  two  numbers  is  5  ;    and  the  antecedent  i?  4 
dollars  :    what  is  the  consequent  ? 

Analysis. — Since  the  ratio  is  equal  to 
the  quotient  of  the  consequent  divided 
by  the  antecedent,   it  follows: 

1st.  That  the  consequent  is  equal  to 
the  antecedent  multiplied  hy   the  ratio : 

2d.  That  the  antecedent  is  equal  to 
the  consequent  divided  hy   the  ratio. 


OPERATIOr. 
Batio 

5     = 


conset  {uent. 


antecedent. 

5  X  ant.  =  cons. 
$4  X  5  =  $20  =  cons. 


Examples. 

1.  The  ratio  of   two   numbers  is    t,    and   the   antecedent   is 
16cwt.  :    what  is  the  consequent? 

2.  The  consequent  is   30  tons,  and  the  ratio  is  6  :    what  is 
the  antecedent  ? 


RATIO   AND    PROPORTION.  209 

3.  The  antecedent  is  15,  and  the  ratio  is  4  :  what  is  the 
consequent  ? 

4.  The  ratio  of  two  numbers  is  1|,  and  the  consequent  is 
7  :   what  is  the  antecedent  ? 

5.  The  ratio  of  two  numbers  is  J,  and  the  antecedent  is  | . 
what  is  the  consequent  ? 

6.  The  ratio  of  the  monthly  wages  of  two  men  is  8  :  the 
greater  wages  of  one  is  $256:  what  is  the  wages  of  the  other? 

7.  The  ratio  is  25,  and  the  consequent  is  14  X  5  x  10  : 
what  is  the  antecedent? 

8.  The  value  of  a  horse  is  2J  times  that  of  an  ox  :  the 
yalue  of  the  horse   is  $143  :   what  is   the   value   of  the   ox  ? 

SIMPLE    PROPORTION. 

237.  A  Simple  Proportion  is  an  expression  of  equality 
between  two  simple  and  equal  ratios.    Thus,  the  two  couplets, 

4  :  20  and  1  :  5, 
having  the  same  ratio  6,  form  a  proportion,  and  are  written, 

4     :     20     :  :     1     :     5, 

by  simply  placing  a   double   colon  between  the  couplets.     The 

terms  are  read, 

4  is  to  20    as     1  is  to  5, 

and  taken  together,  they  are  called  a  proportion. 

238.  The  1st  and  4th  terms  of  a  proportion  are  called  the 
extremes;  the  2d  and  3d  terms,  the  means.  Thus,  in  the  pro- 
portion, 

6     :     24     :  :     8     :     32, 

6  and  32  arc  the  extremes^  and  24  and  8  the  means : 

_,.  24         32 

Smce  ^   =    8" 

WQ  shall  have,  by  reducing  to  a  common  denominator, 
24  X  8  _  32  X  6 
6X8    ~    6X8 


210  RATIO   AND    PROPORTION. 

But  since  the  fractions  are  equal,  and  have  the  same  de- 
nominators, their  numerators  must  be  equal,  viz.  : 

24  X  8  =  32  X  6  ;   that  is, 

In  any  proportion,  the  product  of  the  extremes  is  equal  to 
the  product  of  the  means. 

Thus,  in   the  proportions,  j 

1     :       8     :  :     2     :     16  ;    we  have  1  x  16  =    2x8. 
4     :     12     :   :     8     :     24  ;     "       "      4  X  24  =  12  X  8. 
239.    Since,  in  any  proportion,  the  product  of  the  extremes 
is  equal  to  the  product  of  the  means,  it  follows  that, 

1st.  If  the  product  of  the  means  he  divided  by  one  of  the 
extremes,  the  quotient  will  he  the  other  extreme. 

Thus,  in  the  proportion, 

4     :     16     :   :     6     :     24,  and  4  x  24  =  16  x  6  =  96 ; 

then,  if  96,  the  product   of  the   means,  be  divided  by  one   of 

the  extremes,   4,  the  quotient  will  be  the  other  extreme,   24  ; 

or,  if  the  product  be  divided  by  24,  the  quotient  will   be   4. 

2d.  If  the  product  of  the  extremes  he  divided  hy  either  of 
the  means,  the  quotient  will  he  the  other  mean. 

Thus,  if  4  X  24  =  16  X  6  =  96  be  divided  by  16,  the 
quotient  will  be  6  ;  or  if  it  be  divided  by  6,  the  quotient 
will  be   16. 

Note. — Wc  shall  denote  the  required  term  of  a  proportion  by 
the  letter  x. 


235.  When  the  antecedent  is  less  than  the  consequent,  what  does 
the  ratio  express?  What  does  it  express  when  the  antecedent  is 
greater  than  the  consequent  ? — 336.  To  what  is  the  consequent  e(iual, 
in  any  ratio?  To  what  is  the  antecedent  equal? — 237.  What  is  a 
simple  proportion? — 238.  Which  are  the  extremes  of  a  proportion? 
Which  the  means?  What  is  the  product  of  the  means  equal  to? — 
239,  If  the  product  of  the  means  be  divided  by  one  of  the  extremes, 
what  is  the  quotient?  If  the  product  of  the  extremes  be  divided  by 
one  of  the  means,  what  is  the  quotient? 


KATIO  AND    FKOPORTIOX. 


211 


Examples. 
Find  the  required  term  in  each  of  the  following  examples  : 


5 

30 

9 

X 

5 

$45 

i 

nV 

10 
12 

X 
X 


X. 

36. 
27. 
^^• 

5,   10,  ami   lii 


The  first  three 'terras  of  a  proportion  are 
vhat  is  the  fourth  terra? 

G.   The  first  three  terms  of  a  proportion  are  6,  24,  and   14 
what  is  the  fourth  term? 

7.  The  first,  second,  and  fourth  terms  of  a  proportion  are  W, 
12,  and   16:   what  is  the  third  term? 

8.  The  first,  third,  and  fourth  terms  of  a  proportion  are  1 6, 
8,  and  20  :   what  is  the  second  term  ? 


DIRECT    AND    INVERSE    PROPORTION. 

240.  It  often  happens,  that  two  numbers  which  are  cora- 
})ured  toj^ethcr,  may  undergo  certain  changes  of  value,  in  whicii 
case  they  represent  variable  and  not  Jixed  quantities.  Thus, 
when  we  say  that  the  amount  of  work  done,  in  a  single  day, 
will  be  proportional  to  the  number  of  men  employed,  we  mean, 
that  if  we  increase  the  number  of  men,  the  amount  of  work 
done  will  also  be  increased;  or,  if  we  diminish  the  number  of 
men,  the  work  done  will  also  be  diminished.  This  is  called 
Dii-ect  Proportion. 

If  we  say  that  a  barrel  of  flour  will  serve  12  men  a  certain 
time,  and  ask  how  long  it  will  serve  24  men,  the  time  will  be 
less :  that  is,  the  time  will  decrease  as  the  number  of  mea 
is  increased,  and  will  increase  as  the  number  of  men  is  d^- 
creased.    This  is  called,  Inverse  Proportion;  hence, 

1.  Two  numbers  are  directly  projoortional,  ichen  they  in- 
crease or  decrease  together;  in  which  case  their  ratio  is 
always  the  same. 


k 


212  RATIO   AND    PItOPORTlON. 

2.  Two  numbers  are  inversely  proportional,  when  one 
increases  as  the  other  decreases ;  in  which  case  their  product 
is  always  the  same. 

Note. — This  is  sometimes  called,  Reciprocal  Proportion. 

First    Illustration. 

If  we  refer  to  the  numeration  table  of  integral  and  decimal  num 
oers  (Art.  190),  we  see  that*  the  unit  of  the  first  place,  at  the  left 
of  1,  is  1  ten;  that  is,  a  number  ten  times  as  great  as  1.  Tlie  unit 
of  the  first  decimal  place  at  the  right,  is  1  tenth,  a  number  on]v 
one-tenth  of  1.  The  unit  of  the  second  place,  at  the  left,  is  one 
hundred  times  as  great  as  1 ;  while  the  unit  of  the  second  place, 
at  the  right,  is  only  one  hundredth  of  1 ;  and  similarly  for  all 
other  corresponding  places.     Hence, 

The  units  of  place,  taken  at  equal  distances  from  the  m4t 
1,  are  inversely  proportional. 

Second  Illustration. 

The  floor  of  a  room  is  20  feet  long:  what  must  be  its  bread'Ji 
in  order  that  it  may  contain  360  square  feet? 

Analysis. — The  length  of  the  floor,  operation. 

multiplied  by  its  breadth,  will  give  the  360  _ 

area  or  contents;   hence,   the  area,  di-  20  ~  18  it.  Dreaclia. 

vided    by    the    length,    will    give    the 

breadth.  If  the  contents  remain  the  same,  the  length  will  incrodse 
as  the  breadth  diminishes;  and  the  reverse.  Hence,  when  the  con- 
tents are  the  same^  the  length  and  breadth  are  inversely  proportional. 

COMPOUND   PEOPOETION. 

241.  A  Compound  Proportion  is  the  comparison  of  the 
erras  of  two  equal  ratios,  when  one  or  both  are  compound. 

Thus,     ^l        :      ^l      ::     5       :      6; 


Or. 


:(  ■■  i\  ■■  i\  ■■  :i 


RATIO  AND    PROPORTION.  213 

Any  compound  proportion  may  be  reduced  to  a  simple  one, 

by  multiplying  the   elements   of  each   term  together  ;   thus,  by 

multiplying   together   the   elements   of  the   last  proportion,   we 

have, 

20     r     24     :  :     15     :     18. 

Dcnce,  in  any  compound  proportion, 

^Tlie  product  of  the  extremes  is  equal  to  the  product  of  the 
means  ;   and  the  r-equired  term  may  he  found  as  in  Art.  239. 

What  are  the  required  terms  in  the  following  proportions  ? 

1.  3    X    9     :     12    X    6     :  :     15     :     a:. 

2.  5    X    9     :     10    X    9     :  :     18     :     a:. 

242.  If  an  element  is  unknown,  denote  it  by  x.  Then,  if 
all  the  parts  arc  known  except  one  element,  as  in  the  follow- 
ing proportion, 


I      =       ^cl      ==        ?|      =      l\ 


that  element  is  equal  to  the  product  of  the  means  divided  by 
the  product  of  all  the  elements  of  the  first  term  and  the  known 
elemerds  of  the  fourth  term  ;  thus, 

5x6x3xT 
2X  21  X5 

1.   What  is  the  required  clement  in  the  proportion, 


3J  5J  1 

2^        :         3>        ::         3 
5)  8)  2 


'I 

X) 


240.  When  are  two  numbers  directly  proportional?  When  are  two 
numbers  inversely  proportional  ?  What  is  then  said  of  their  product  ? 
Give  the  first  illustration  of  inverse  proportion..  Give  the  second. — 
241.  What  is  a  compound  proportion?  What  is  the  product  of  the 
extromea  equal  to? — 242.   ITow  do  you  find  the  unknown  element? 


214  SINGLE   RULE  OF   THREE. 


SINGLE    BULE    OF    THREE. 

243.  The  Single  Rule  of  Three  is  the  process  of  finding 
from  three  given  numbers,  a  fourth,  to  which  one  of  them 
sliall  have  the  same  ratio  as  exists  between  the  other  two. 

1.  If  8  barrels  of  flour  cost  $56,  what  will  9  barrels  cost, 
at  the  same  rate. 

Note. — We  shall  denote  the  required  term  of  the  proportion  by 
the  letter  x. 

Analysis. — The   condition,  "at  the  statement. 

same  rate,"  requires  that  the  quantity^       lar.  tar.       $        $ 

8  barrels  of  flour,  have  the  same  ratio  8   :  9   :  :  56   :  a?, 

to  the  quantity,  9  barrels,  as  $56,  the  56  X  9 
cost  of  8  barrels,  to  x  dollars,  the  cost  3        ~  ^ 

of  9  barrels.  ' 

Note. — It  is  plain  that  8  barrels  of  flour  will  cost  less  than  9  bar- 
rels :  hence,  the  3d  term  is  less  than  the  4th,  and  these  terms  are 
directly  proportional. 

2.  If  36  men,  in  12  days,  can  do  a  certain  work,  in  what 
time  will  48  men  do  the  same  work? 

Analysis. — Write    the    required  operation. 

term,  tc,  in  the  4th  place,  and  the         48   :  36   :  :  12   :  a?, 
term  12,  having  the  same  unit  value,  36  X  12 

in  the  third  place.  ^  =       43       =  ^  days. 

Then,  analyzing  the  question,  we 
see  that  48  meu  will  do  the  work  in  a  less  time  than  86  Ta<in ; 
therefore,  the  3d  term  will  be  greater  than  the  4th,  which  requires 
that  the  1st  shall  be  greater  than  the  2d.  This  brings  48  men 
into  the  first  place,  and  36  men  into  the  2d.  The  reason  of  this 
is  obvious:  for,  the  work  done  by  36  men  in  12  days,  is  a  Jixed 
quantity.  If  a  less  number  of  men  are  employed,  it  will  require 
liore  time  to  do  the  work:  if  a  greater  number  are  employed,  it 
will  require  less  time ;  hence,  the  men  and  time  are  inversely  'pro- 
portional. 

243,  WHiat  ik  the  sin/rlo  rule  of  tlireo?    (Uvo  tin  rule. 


8INGLK    iiri.K    OF   THREE.  215 

Kule  -I.  Wi'ile  the  required  term,  x,  in  the  ith  place, 
and  the  term  having  the  same  unit  value  in  the  Zd  place: 

II.  Then  analyze  the  question,  and  see  whether  the  fourth 
term  is  greater  or  less  than  the  third;  ivhen  greater,  write 
the  least  of  the  remaining  terms  in  the  Jirst  place,  and  when 
less,  icrite  the  greater  there,  and  the  remaining  term  in  the 
second  place : 

III.  ITien  multiply  the  second  and  third  terms  together^ 
and  divide  the  product  by  the  Jirst. 

This  rule  gives,  when  quantity  and  cost  are  considered ; 

quantity     :     quantity     :  :     cost     :     cost. 
When  labor  and  time  are  considered, 

labor     :    labor     :  :     tune     :    time. 

When  labor  and  work  done  are  considered, 

labor     :     labor     :  :     work  done     :     work  done. 

Notes. — 1.  If  tho  first  and  second  terms  have  different  units,  they 
must  bo  reduced  to  tho  same  unit. 

2.  If  the  third  term  is  a  compound  denominate  member,  it  must  be 
reduced  to  the  smallest  unit. 

3.  The  preparation  of  the  terms,  and  writing  them  in  their  proper 
places,  is  called  the  statement. 

4.  When  the  unknown  term  and  tho  term  named  in  connection 
with  it,  form  the  extremes,  the  proportion  between  them  is  Inverse. 

Examples. 

1.  If  8  hats  cost  $24,  what  will  110  hats  cost,  at  the  same 
ate? 

2.  If  2  barrels  of  flour  cost  $15,  what  will  12  baiTels  cost? 

3.  If  I  walk  168  miles  in  6  days,  how  far  can  I  walk,  at 
the  same  rate,  in  18  days? 

4.  If  8  lb.  of  sugar  cost  $1.28,  how  much  will  13  lb.  cost  ? 

5.  If  300  barrels  of  flour  cost  $2100,  what  will  125  barrels 
coijt  ? 


21(>  SINGLE    RULPJ   OF  THREE. 

6.  If  120  sheep  yield  330  pounds  of  wool,  bow  many  pounds 
will  36  sheep  yield? 

I.  If  80  yards  of  cloth  cost  $340,  what  will  650  yards  cost  ? 

8.  What  is  the  value  of  4  cwt.  of  sugar,  at  5  cents  a  pound  ? 

9.  If  6  gallons  of  molasses  cost  $1.95,  what  will  6  hogs- 
heads cost? 

10.  If  16  men  consume  560  pounds  of  bread  in  a  month 
haw  much  will  40  men  consume? 

II.  If  a  man  travels  at  the  rate  of  630  miles  in  12  days, 
how  far  will  he  travel  in  a  leap  year,  Sundays  excepted  ? 

12.  If  2  yards  of  cloth  cost  13.25,  what  will  be  the  cost 
of  3  pieces,  each  containing  25  yards  ? 

13.  If  3  yards  of  cloth  cost  18s.  New  York  currency,  what 
will  36  yards  cost  ? 

14.  If  it  requires  eight  shillings  and  four  pence  to  buy  eight 
ounces  of  laudanum,  how  many  ounces  can  be  purchased  for 
Is.  6d.  ? 

15.  If  5  A.  IR.  16  P.  of  land,  cost  $150.5,  what  will 
125 A.  2R.  20 P.  cost? 

16.  If  13  cwt.  2qr.  of  sugar  cost  $129.93,  what  will  be  the 
cost  of  9  cwt.  ? 

It.  The  clothing  of  a  regiment  of  750  men  cost  iE2834  5s.: 
what  will  it  cost  to  clothe  a  regiment  of  10500  men? 

18.  If  3 J  yards  of  cloth  will  make  a  coat  and  vest,  when 
the  cloth  is  IJ  yards  wide,  how  much  cloth  will  be  needed 
when  it  is  |-  of  a  yard  in  width  ? 

19.  If  I  have  a  piece  of  land  16|-  rods  long  and  3 J  rods 
wide,  what  is  the  length  of  another  piece  that  is  t  rods  wide 
and  contains  an  equal  area  ?  « 

20.  How  many  yards  of  carpeting  that  is  three-fourths  of 
yard   wide,  will    carpet    a   room  36   feet  long   and   30  feet  in 
breadth  ? 

21.  If  a  man  can  perform   a  journey  in   8  days,  walking 
hours  a  day,  how  many  days    will    it   require   if  he  walks   10 
hours  a  day  ? 


APPLICATIONS.  217 

22.  If  a  family  of  15  persons  liave  provisions  for  8  months, 
Ijy  bow  many  must  the  family  be  diminished  that  the  provisions 
may  last  2  years? 

23.  A  garrison  of  4600  men  has  provisions  for  G  months  :  to 
what  number  must  the  garrison  be  dimmished  that  the  pro- 
visions may  last  2  years  and  6  months? 

24.  A  certain  amount  of  provisions  will  subsist  an  army  o* 
9000  men  for  90  days :  if  the  army  be  increased  by  GO 00, 
DOW  long  will  the  same  provisions  subsist  it? 

25.  If  3  yd.  2qr.  of  cloth  cost  $15.t5,  how  much  will  8  yd. 
3qr.  of  the  same  cloth  cost? 

26.  If  .5  of  a  house  cost  $201.5,  what  will  .95  cost? 

21.  What  will  26.25  bushels  of  wheat  cost,  if  3.5  bushels 
cost  18.40? 

28.  If  the  transportation  of  2.5  tons  of  goods  2.8  miles 
costs  $1.80,  what  is  that  per  cwt.  ? 

29.  If  f  of  a  yard  of  cloth  cost  $2.16,  what  will  be  the 
cost  of  5^  pieces,  each  containing  44 T  yards  ? 

30.  If  f  of  an  ounce  cost  $-}J,  what  will  l|oz.  cost? 

31.  What  will  be  the  cost  of  16|lb.  of  sugar,  if  14|ilb. 
cost  $lf  ? 

32.  If  $19i-  will  buy  14  J  yards  of  cloth,  how  much  will 
89f  yards  cost  ? 

33.  If  |-  of  a  barrel  of  cider  cost  ^j  of  a  dollar,  what  will 
]^  of  a  baiTcl  cost? 

34.  If  y\  of  a  ship  cost  $2880,  what  will  Jf  of  her  cost? 

35.  What  will  1161  yards  of  cloth  cost,  if  462  yards  cost 
$150.66? 

36.  If  6  men  and  3  boys  can  do  a  piece  of  work  in  330 
days,  how  long  will  it  take  9  men  and  4  boys  to  do  the 
same  work,  under  the  supposition  that  each  boy  does  half  as 
much  as  a  man? 

37.  If  4  men  can  do  a  piece  of  work  in  80  days,  how 
many  days  will   16   men  require  to   do   the  same  work? 

10 


21$  SINGLE   nULE    OF   THREE. 

38.  If  21  sappers  make  a  trench  in  18  clays,  how  many 
days    will    7    men   require   to   make   a   similar   trench  ? 

39.  A  certain  piece  of  grass  was  to  be  mowed  by  20  men 
in  6  days ;  one-half  the  workmen  being  called  away,  it  is 
required  to  find  in  what  time  the  remainder  will  complete 
the  work  ? 

40.  If  a  field  of  gram  be  cut  by  10  men  in  12  days, 
in  liow  many  days  would  20  men  have  cut  it  ? 

41.  If  90  barrels  of  flour  will  subsist  100  men  for  120 
days,  how  long  will  they  subsist  75  ? 

42.  If  a  traveller  perform  a  journey  in  35.5  days,  when  the 
days  are  13.566  hours  long,  in  how  many  days  of  11.9  hours, 
will  he  perform  the  same  journey  ? 

43.  If  50  persons  consume  600  bushels  of  wheat  in  a  year, 
how  long  would  they  last  5  persons  I 

44.  A  certain  work  can  be  done  in  12  days,  by  working 
4  hours  each  day  :  how  many  days  would  it  require  to  do 
the   same   work,   by   working  9    hours   a   day  ? 

45.  If  1j\  barrels  of  fish  cost  $31J,  what  will  32J  bar- 
rels   cost  ? 

46.  How  much  wheat  can  be  bought  for  $96|-,  if  2  bu. 
Ipk.   cost   $1.93-1? 

47.  If  I  of  a  yard  of  cloth  cost  8 If,  what  will  7-J 
yards  cost  ? 

48.  What  will  be  the  cost  of  37.05  square  yards  of  pave- 
ment, if  47.5   yards   cost   $72.25  ? 

49.  If  3  paces  or  common  steps  be  equal  to  2  yards, 
how   many  yards   will    160    paces   make  ? 

60.  If  a  person  pays  half  a  guinea  a  week  for  his  board, 
how  long   can  he   board  for   ^£21? 

51.  If  12  dozen  copies  of  a  certain  book  cost  $54.72 
what   will   297    copies   cost   at   the   same   rate  ? 

52.  If  an  army  of  900  men  require  $3618  worth  of  pro- 
visions  for   90   days,    what    will   be    the    cost    of    subsistenoe. 


AITLICATIONS.  219 

for  the    same    time,    when    the    aiiny    is    increased    to    4500 
men  ? 

53.  A  grocer  bouglit  a  hogshead  of  rum  for  80  cents  a 
gallon,  and  after  adding  water  sold  it  for  60  cents  a  gallon, 
when  he  found  that  the  selling  and  buying  prices  were  pro- 
portional to  the  original  quantity  and  the  mixture  :  how  much 
water  did   he   add? 

54.  A  man  failing  in  business,  pays  60  cents  for  every 
dollar  which  he  owes  ;  he  owes  A  13570,  and  B  $1875 : 
how  much  docs  he  pay  to  each? 

55.  A  bankrupt's  effects  amount  to  $2328.75,  his  debts 
amount  to  $3726  :  what  will  his  creditors  receive  on  a  dol- 
lar? 

56.  If  a  person  drinks  80  bottles  of  wine  in  3  months 
of  30   days  each,  how   much   does    he   drink   in   a   week  ? 

57.  If  4f  yards  of  cloth  cost  14s.  8d.  New  York  currency, 
what  will    40A   yards  cost? 

58.  If  a  grocer  uses  a  false  balance,  giving  only  14'  oz. 
for  a  pound,  how  much  will  154|-lb.  of  just  weight  give, 
when   weighed  by   the   false  balance? 

59.  If  a  dealer  in  liquors  uses  a  gallon  measure  which  is 
too  small  by  ^  of  a  pint,  what  will  be  the  true  measure  of 
100  of  the  false  gallons  ? 

00.  After  A  has  travelled  96  miles  on  a  journey,  B  sets 
out  to  overtake  him,  and  travels  23  miles  as  often  as  A 
travels  19  miles  :  how  far  will  B  travel  before  he  over- 
takes A  ? 

61.  A  person  owning  |-  of  a  coal  mine,  sold  |  of  his 
share   for   $9345  :    what   was   the   value   of    the   whole  mine  ? 

62.  At   what   time,    between    6    and    7    o'clock,    will    the 
our   and   minute   hands   of  a   clock   be   exactly   together  ? 

63.  If  a  staff,  5  feet  long,  casts  a  shadow  of  7  feet,  what 
is  the  height  of  a  steeple,  whose  shadow  is  196  feet,  at  the 
same   time   of  day? 


220  SINGLE  RULE  OF  THREE. 

64.  A  can  do  a  piece  of  work  in  3  days,  B  in  4  days, 
and  C  in  6  days  :  in  what  time  will  they  do  it,  working 
together  ? 

65.  A  can  build  a  wall  in  15  days,  but  with  the  assistance 
of  C,  he  can  do  it  m  9  days  :  in  what  time  can  C  do  it 
alone  ? 

66.  If  120  men  can  build  J  mile  of  wall  in  151  days,  how 
many  men  would  it  require  to  build  the  same  wall  in  40|  days  ? 

61.  If  3  horses,  or  5  colts,  eat  a  certain  quantity  of  oats  in 
40  days,  in  what  time  will  t  horses  and  3  colts  consume  the 
same  quantity? 

68.  If  a  person  can  perform  a  journey  in  24  days  of  10 J 
hours  each,  in  what  time  can  he  perform  the  same  journey, 
when  the  days  are  12 J  hours  long? 

69.  A  piece  of  land,  40  rods  long  and  4  rods  wide,  is  equiv- 
alent to  an  acre  :  what  is  the  breadth  of  a  piece  15  rods  long 
that  is  equivalent  to  an  acre? 

70.  If  a  person  travelling  12  hours  a  day  finishes  one-half  of 
a  journey  in  ten  days,  in  what  time  will  he  finish  the  remain- 
ing half,  travelling  9  hours  a  day? 

11.  How  many  pounds  weight  can  be  carried  20  miles,  for 
the  same  money  that  4^  cwt.  can  be  earned  36  miles  ? 

12.'  If  12  liorses  eat  a  certain  quantity  of  hay  in  1J  weeks, 
how  many  horses  will  consume  the  same  in  90  weeks  ? 

13.  A  watch,  which  is  10  minutes  too  fast  at  12  o'clock,  on 
Monday,  gains  3  min.  10  sec.  per  day:  what  will  be  the  time, 
by  the  watch,  at  a  quarter-past  ten  in  the  morning  of  the  fol- 
lowing Saturday? 

74.  Two  persons,  A  and  B,  are  on  the  opposite  sides  of  a 
wood,  which  is  536  yards  in  circumference ;  they  begin  to 
travel  in  the  same  direction  at  the  same  moment ;  A  goes  at 
the  rate  of  11  yards  per  minute,  and  B  at  the  rate  of  34 
yards  in  3  minutes  :  how  many  times  must  A  go  round  the 
wood  before  he  is  overtaken  by  B  ? 


DOUBLE    RULE    OF  THREE. 


221 


DOUBLE    RULE    OF    THREE. 

244.   The  Double  Rule   of  Three  is  an  application   of  tht 
principles  of  Compound  Proportion. 

1.    If   8   men   iu    12  days  can   build   80   rods   of  wall,    how 
iiuch  will  6  men  build  in  18  days? 


8 
12 


STATEMENT. 
6 

18 


}      ■■     d\     ': 


80 


OPERATION. 
18    X    6    X    80 

12x8 


Analysis.  —  We  write  the 
required  term  in  the  4th 
place,  and  the  80  rods  in  the 
3d.  Then,  since  the  wall  built 
IS  directly   proportional   to  the 

number  of    men  multiplied   by  x  =:  ^^ ,^^  ^  ^  ^^  _  qq  I'ods. 

the  number  of  days,  6  x  18  is 
written  in  the  second  place, 
and  the  remaining  term  in  the  first  place. 

2.  If  20  men  can  perform  a  piece  of  work  in  12  days,  work- 
ing 9  hours  a  day  (that  is,  in  108  hours),  how  many  men 
will  accomplish  the  same  work  in  6  days,  working  10  hours  a 
day  (that  is,  in  60  hours)? 


8TATKMENT. 


[  -'11  - 


20 


X. 


X  = 


OPKRATION. 

12  x  9  X  20 

Tx  10 


=  36  men. 


Analysis. — Write  the  required 
term,  x,  in  the  4th  place,  and  20 
men,  having  the  same  unit,  in 
the  3d  place.  Since  20  men 
require  108  hours  to  do  the  work, 
more  men  will  be  required  to  do 
the  same  work  in  60  hours ; 
therefore,  tlie  terms  named  in  connection  with  each  other,  are 
inversely  proportional:  hence,  6  X  10  =  60,  must  be  written  in  the 
first  place. 

3.  If  24  men,  in  6  days,  working  t  hour?  a  day,  can  build 
a  wall  115  feet  long,  3  feet  thick,  and  4  feei  ^igh,  how  long 
a  wall  can  36  men  build  in  12  days,  working  14  hours  a  day, 
if  the  wall  is  4  feet  thick  and  5  feet  high. 


244.   What  is  the  double  rule  of  three? 


222  DOUBLE  RULE  OF  THREE. 


Analysis.— In  this 

STATEMENT. 

example,  an  element, 

24) 

3C  )             115)          x) 

viz.,    length    of  wall 

6  - 

:     12        ::         3        :     4[ 

is  required.     This  el- 

0 

U)                 i)          5 ) 

ement,  denoted  by  cc, 

OPERATION. 

is    put    in     the    4th 
place  with  the  other 

36 

X  12  X  14  X  115  X  3  X  4 

Jb    

24x6x1x4x5 

elements     composing 

the   4th   term. 

=  414 


Rule. 

I.  Write  the  required  term,  or  the  term  containing  the  re- 
quired element,  in  the  Uh  place,  and  the  term  having  the 
same  unit  value  in  the  third  place : 

II.  Then  analyze  the  question,  and  see  whether  the  terms 
named  in  connection  with  each  other  are  directly  or  inversely 
proportional :  when  directly  proportional,  write  the  term  named 
in  connection  with  the  4th  term,  in  the  2d  place  ;  and  when 
inversely,  ivrite  it  in  the  first  place:  then  fi)id  the  required 
term  or  clement  (Art.  242). 

Examples. 

1.  If  2  men  can  dig  125  rods  of  ditch  in  75  days,  in  how 
many  days  can  18  men  dig  243  rods  ? 

2.  If  400  soldiers  consume  5  barrels  of  flour  in  12  days, 
how  many  soldiers  will  consume  15  barrels  in  2  days  ? 

8.  If  a  person  can  travel  120  miles  in  12  days  of  8  hours 
each,  how  far  will  he  travel  in  15  days  of  10  hours  each  ? 

4.  If  a  pasture  of  16  acres  will  feed  6  horses  for  4  months 
Low  many  acres  will  feed  12  horses  for  9  months? 

5.  If  60  bu.^uels  of  oats  will  feed  24  horses  40  days,  how 
long  will  30  bushels  feed  48  horses  ? 

6.  If  82  men  build  a  wall  36  feet  long,  8  feet  high,  und  4 
feet  thick,  in  4  days;  in  what  time  will  48  men  build  a  wail 
864  feet  long,  6  feet  high,  and  3  feet  wide  ? 


APPLICATIONS.  223 

7.  If  the  freiglit  of  80  tierces  of  sugar,  each  weighiug  3J 
hundredweight,  for  150  miles,  is  $84,  what  must  be  paid 
for  tlie  freight  of  30  hogsheads  of  sugar,  each  weighing  12 
hundredweight,  for  50  miles  ? 

8.  A  family  consisting  of  G  persons,  usually  drink  15.6  gal- 
ons    of   beer   in   a  week  :    how  much  will   they  drink  in   12.5 

weeks,  if  the  number  be  increased  to  9  ? 

9.  If  12  tailors  in  7  days  can  finish  14  suits  of  clothes, 
how  many  tailors  in  19  days  can  finish  the  clothes  of  a  regi- 
ment of  494  men  ? 

10.  If  a  garrison  of  3600  men  eat  a  certam  quantity  of 
bread  in  35  days,  at  24  ounces  per  day  to  each  man,  how 
many  men,  at  the  rate  of  14  ounces  per  day,  will  eat  twice  as 
much  in  45  days  ? 

11.  A  company  of  100  men  drank  iB20  worth  of  wine  at 
2s.  6d.  per  bottle  :  how  many  men,  at  the  same  rate,  will  £1 
worth  supply,  when  wine  is  worth  Is.  9d.  per  bottle? 

12.  A  garrison  of  3600  men  has  just  bread  enough  to  allow 
24  oz.  a  day  to  each  man  for  34  days  ;  but  a  siege  coming  on, 
the  garrison  was  reinforced  to  the  number  of  4800  men  :  how 
many  ounces  of  bread  a  day  must  each  man  be  allowed,  to 
hold  out  45  days  again«t  the  enemy? 

13.  Bought  5000  planks,  15  feet  long  and  2|  inches  thick  ; 
how  many  planks  are  they  equivalent  to,  of  12^  feet  long  and 
1  f  inches  thick  ? 

14.  If  12  pieces  of  cannon,  eighteen-pounders,  can  batter 
down  a  castle  in  3  hours,  in  what  time  would  nine  twenty-four- 
pounders  batter  down  the  same  castle,  both  pieces  of  cannon 
bemg  fired  the  same  number  of  tunes,  and  their  balls  flying 
with  the  same  Telocity  ? 

15.  If  the  wages  of  13  men  for  7 J  days,  be  $149.76,  what 
will  be  the  wages  of  20  men  for  15J  days? 

16.  If  a  footman  travel  264  miles  m  6f  days  of   12 J  hours 


224        DOUBLE  rulp:  of  three. 

each,  in  how  many  days  of  lOf  hours  each  will  he  travel  129f 
miles  ? 

It.  If  120  men  in  3  days,  of  12  hours  each,  can  dig  a 
a  trench  of  30  yards  long,  2  feet  broad,  and  4  feet  deep,  how 
many  men  would  be  required  to  dig  a  trench,  50  yards  long, 
()  feet  deep,  and  IJ  yards  broad,  in  9  days  of  15  hours  each? 

18.  If  a  stream  of  water  running  into  a  pond  of  115  acres, 
raises  it  10  inches  in  15  hours,  how  much  would  a  pond  of  80 
acres  be  raised  by  the  same  stream  in  9  hours? 

19.  A  person  having  a  journey  of  500  miles  to  perform, 
walks  200  miles  in  8  days,  walking  12  hours  a  day  :  in  how 
many  days,  walking  10  hours  a  day,  will  he  complete  the  re- 
mainder of  the  journey  ? 

20.  If  1000  men,  besieged  in  a  town,  with  provisions  for  28 
days,  at  the  rate  of  18  ounces  per  day  for  each  man,  be  rein- 
forced by  600  men,  how  many  ounces  a  day  must  each  man 
have  that  the  provisions  may  last  them  42  days  ? 

21.  If  a  bar  of  iron  5  ft.  long,  2  J  in.  wide,  and  If  in.  thick, 
weigh  451b.,  how  much  will  a  bar  of  the  same  metal  weigh 
that  is  t  ft.  long,  3  in.  wide,  and  2^  in.  thick  ? 

22.  If  5  compositors  in  16  days,  working  14  hours  a  day, 
can  compose  20  sheets  of  24  pages  each,  50  lines  in  a  page, 
and  40  letters  in  a  line,  in  how  many  days,  working  T  hours  a 
day,  can  10  compositors  compose  40  sheets  of  16  pages  in  a 
sheet,  60  lines  in  a  page,  and  50  letters  in  a  line? 

23.  Fifty  thousand  bricks  are  to  be  removed  a  given  dis- 
tance in  10  days.  Twelve  horses  can  remove  18000  in  6  days  : 
how  many  horses  can  remove  the  remainder  in  4  days? 

24.  If  248  men,  in  5 J  days  of  11  hours  each,  dig  a  trench 
of  t  degrees  of  hardness,  232 1  yards  long,  of  wide,  and  2 J 
deep,  in  how  many  days,  of  9  hours  long,  will  24  men  dig  a 
trench  of  4  degrees  of  hardness,  33tJ  yarc's  long,  5f  wide,  and 
3i   deep? 


PARTNERSHIP.  225 


PARTNERSHIP. 

245.  A  Partnership,  or  Firm,  is  an  association  of  two  or 
more  persons,  under  an  agreement  to  share  the  profits  and 
'osscs  of  business. 

Partners  arc  the  persons  thus  associated. 

246.  Capitai,  or  Stock,  is  the  amount  of  money  or  prop- 
erty contributed  by  the  partners,  and  used  in  the  business. 

Profit  is  the  increase  of  capital  between  two  given  dates. 
Loss  is  the  decrease  of  capital  between  two  given  dates. 
Dividend  is  the  amount  of  profit  ajiportioned  to  each  partner. 

247.  Assets  of  a  Firm,  are  its  cash  on  hand,  property,  and 
all  debts  due  to  it. 

248.  Liabilities  of  a  Firm,  embrace  all  the  debts  wliich  it 
owes,  and  all  its  indorsements. 

249.  Solvency  is  when  the  assets  exceed  the  liabilities. 

250.  Insolvency  is  when  the  liabilities  exceed  the  assets. 
25  L  An  Assignment  is  a  transfer   of  the   assets  of   an    in- 
solvent person  or  firm  to  others,  for  the  benefit  of  creditors. 

252.  Assignees  are  the  persons  to  whom  such  transfer  is 
made. 

253.  When  the  capital  of  each  partner  is  employed  for  tha 
same  time. 

Since  the  profit  arises  from  the  use  of  the  capital,  each 
man's  share  of  it  should  be  proportional  to  his  amount  of 
stock.     Hence, 

245.  What  is  a  partnership  or  firm?  Wliat  are  partners? — 246.  Wliat 
Is  canitiil  or  stock?  What  is  profit?  What  is  loss?  What  is  a  div- 
idend?—247.  What  arc  assets?— 248.  What  arc  liabilities ?— 249.  What 
is  solvency? — 250.  What  is  insolvency? — 2ol.  What  is  an  assignment? 
252.  What  are  assignees  ?— 253.  What  is  the  rule,  when  each  man's 
capital  is  employed  for  the  same  time? 

io* 


226  PARTNERSHIP. 

Rule. — As  the  wHiole  stock  is  to  each  man^s  stock,  so  is 
the  whole  gain  or  loss  to  each  mail's  share  of  the  gain  or  loss. 

Examples. 

1.  Mr.  Jones  and  Mr.  Wilson  form  a  copartnership,  the 
former  putting  in  $1250,  and  the  latter  $750  :  at  the  end  oi 
the  year  there  is  a  profit  of  $720:  what  is  the  share  of  each? 

1250 

750  STATKMENT. 


2000     :     1250     :  :     720     :     x  =  Jones'  share  =  $450. 
2000     :       750     :  :     720     :     x  =  Wilson's  share  =  $270. 


OPERATION. 


25  18 

lt^0   X  '^'^0 


^000 
^0 


=  x=  $450. 


15  18 

— ^00-  =  ^=^^^'^• 
^0 


2.  A,  B,  and  C,  entered  into  partnership  with  a  capital  of 
$7500,  of  which  A  put  in  $2500,  B  put  in  $3000,  and  C  put 
in  the  remainder  ;  at  the  end  of  the  year  their  gain  was  $3000 : 
what  was  each  one's  share  of  it  ? 

3.  A  and  B  have  a  joint  stock  of  $4200,  of  which  A  owns 
$3600,  and  B,  $600  ;  they  gain,  in  one  year,  $2000  :  what  is 
each  one's  share  of  the  profits  ? 

4.  A,  B,  C,  and  D,  have  $40000  in  trade,  each  an  equal 
share  ;  at  the  end  of  six  months  their  profits  amount  to 
$16000  :  what  is  each  one's  share,  allowing  A  to  receive  $50, 
and  D,  $30,  out  of  the  profits,  for  extra  services  ? 

5.  Three  merchants  loaded  a  vessel  with  flour  ;  A  loaded  50i 
barrels,  B,  700  barrels,  and   C,    1000   barrels  ;    in   a   storm    a 
sea  it  became  necessary  to  throw  overboard  440  barrels  :  what 
was  each  one's  share  of  the  loss  ? 

6.  A  man  bequeathed  his  estate  to  his  four  sons,  in  the  fol- 
lowing manner,  viz. :   to  his  first,  $5000,  to  his  second,  $4500, 


APPLICATIONS.  227 

to  bis  third,  $4500,  aucl  to  his  fourth,  $4000.  But  on  settling 
the  estate,  it  was  found  that  after  paying  the  debts  and  ex- 
penses, only  $12000  remained  to  be  divided:  how  much  should 
each  receive  ? 

7.  A  widow  and  her   two   sons  receive   a  legacy  of  $4500, 
of  wliich  the  widow  is    to   have  ^,  and   the  sons,  each  \.     But 
the  elder  son  dying,  the  whole   is   to   be  divided  in   the   sani 
proportion   between   the   mother   and   younger   son  :   what   will 
each  receive? 

8.  Four  persons  engage  jointly  in  a  land  speculation ;  D 
puts  in   $5499   capital.     They  gain  $15000,  of  which  A  takes 

-1320.50,  B,  $5245.75,  and  C,  $3600.75  :  how  much  capital  did 
A,  B,  and  C  put  in,  and  what  is  D's  share  of  the  gain? 

9.  A  steam-mill,  valued  at  $4300,  was  entirely  destroyed  by 
fire.  A  owned  -J-  oC  it,  B  i,  and  C  the  remainder  ;  supposing 
it  to  have  been  insured  for  $2500,  what  was  each  one's  share 
of  the  loss  ? 

10.  A  copartnership  is  formed  with  a  joint  capital  of 
•^10970.  A  puts  in  $5  as  often  as  B  puts  in  $7,  and  as 
oi'ten  as  C  puts  in  $8  ;  their  annual  gain  is  equal  to  C's  stock  : 
what  is  each  person's  stock  and  gain  ? 

11.  A  man  failing  in  business  is  indebted  to  A,  $475.50, 
to  B,  $362.12^,  to  C,  $250.87J,  and  to  D,  $140.  He  is 
worth  only  $014.25  :  to  how  much  is  each  entitled  ? 

12.  Four  persons.  A,  B,  C,  and  D,  agreed  to  do  a  piece 
of  work  for  $270.  They  were  to  do  the  work  in  the  pro- 
portions of  I,  -J,  J,  and  J  J  :  what  should  each  receive  for  his 
work  ? 

13.  A,  B,  and  C,  form  a  copartnership,  with  a  capital  Oi 
$50000,  of  which  A  puts  in  $18500,  B,  $24650,  and  C,  the 
remainder.  C,  on  account  of  his  superior  knowledge  of  the 
business,  was  to  receive  -^^  of  all  the  profits,  exclusive  of  his 
share.  At  the  end  of  the  year,  tlie  net  profit  is  $7300 ; 
what  should  each  receive  ? 


228  PARTNERSHIP. 

14.  Two  merchants,  A  and  B,  form  a  copartnership.  A 
contributes  $10500,  and  B,  $16500.  At  the  end  of  the  year, 
the  assets  are  $29400,  and  the  liabihties  $4750.  Now,  sup- 
posing the  partnership  to  continue,  with  what  capital  does 
each  partner  commence  the  new  year  ? 

15.  Three  persons  buy  a.  piece  of  land  for  $45G9,  and  the 
>arts  for  which  they  pay  bear  the  following  proportions  to  each 

other,  viz. :  the  sum  of  the  first  and  second,  the  sum  of  the 
first  and  third,  and  the  sum  of  the  second  and  third,  are  to 
each  other  as  i  |,  and  -^q  :  how  much  did  each  pay,  and 
what  part  did  each  own? 

254.     When  the  capital  is  employed    for  unequal  times. 

When  the  partners  employ  their   capital   for   unequal  times, 
the  profits  of  each  will  depend  on  two  circumstances  : 
1st,  On  the  amount  of  capital  he  puts  in  ;   and 
2dly,   On  the  length  of  time  it  is  continued  in  business: 
Therefore,  the  profit  of  each  will  depend   on   the   product  of 
these   two    elements.     The  whole   profit  will  be  proportional  to 
the  sum  of  these  products.     Hence,  the  following 

'Rule.—3Iitltiply  each  man^s  cajntal  by  the  time  he  con- 
tinued it  in  the  firm ;  then  say,  the  sum  of  the  2Jroducts 
is  to  each  product,  so  is  the  whole  gain  or  loss  to  each  man^s 
share. 

Examples. 

1.  A  put  in  trade  $500  for  4  months,  and  B  $600  for  5 
months.     They  gained  $240  :    what  was  the  share  of  each  ? 

OPERATION. 

A^s  cap.     500  X  4  =  2000 
B's  cap.     600  X  5  =  3000 

Sum  of  products  =  5000   :  2000    :  :    240   :  ^  =    $96,  A's 
5000   :  3000    :  :    240   :  :p  =  $144,  B's. 

2.  Three  men  hire  a  pasture  for  $70.20  :  A  put  in  7  horses 


APPLICATIONS  229 

for  3  months  ;  B,  9  horses  for  5  months  ;  and  C,  4  horses 
for  6  months  :    what  part  of  the  rent  should  each  pay  ? 

3.  A  commenced  business  with  a  capital  of  $10000.  Four 
months  afterwards  B  entered  into  partnership  with  him,  and 
put  in  1500  barrels  of  flour.  At  the  close  of  the  year  their 
profits  were  $5100,  of  which  B  was  entitled  to  $2100  :  what 
was  the  value  of  the  flour  per  barrel? 

4  Ou  the  1st  of  January,  1864,  A  commenced  business 
with  a  capital  of  $23000  ;  two  months  afterwards  he  drew  out 
$1800  ;  on  the  1st  of  April,  B  entered  into  partnership  with 
him,  and  put  in  $13500  ;  four  months  afterwards  he  drew  out 
$10000  ;  at  the  end  of  the  year  their  profits  were  $8400  : 
how  much  should  each  receive  ? 

5.  Three  persons  divided  theu*  profits  to  the  amount  of 
$798.  A  put  out  $4000  for  12  months;  B,  $3000  for  15 
mouths  ;  and  C,  $5000  for  8  months  :  to  what  part  of  the 
profits  was  each  entitled? 

6.  Tliree  persons,  C,  D,  and  E,  form  a  copartnership  ;  C's 
stock  is  in  trade  3  months,  and  he  claims  ^^  of  the  gain  ;  D's 
stock  is  in  9  months ;  and  E  put  in  $756  for  4  months,  and 
claims  i  of  the  profits  :   how  much  did  C  and  D  put  in  ? 

7.  Two  persons  form  a  partnership  for  one  year  and  six 
months.  A,  at  first,  put  in  $3000  for  9  months,  and  then 
'$1000  more.  B,  at  first,  put  in  $4000,  and  at  the  end  of 
the  first  year,  $500  more  j  but  at  the  end  of  15  months,  he 
drew  out  $2000.  At  the  end  of  12  months,  C  was  admitted 
as  a  partner  with  $7333^.  The  gain  was  $7400  :  how  much 
should  each  man  receive  ? 

8.  Three  men  take  an  interest  in  a  mining  company.  A 
])ut  in  $480  for  6  months;  B,  a  sum  not  named  for  12 
months  ;  and  C,  $320  for  a  time  not  named :  when  the 
accounts  were  settled,  A  received  $600  for  his  stock  and 
in-ofits  ;  B,  $1200  for  his  ;  and  C,  $520  for'his  :  what  was  B' 
stock,  and  C's  time  ? 

254.  What  is  the  rule  when  the  capital  is  employed  for  unequal  time*. 


:30  PERCENTAGE. 


PERCENTAGE. 

255.  Per  cent,  means  by  the  hundred.  Thus,  1  per  cent, 
of  a  number  is  one-hundredth  of  it  ;  2  per  cent,  is  two-hun- 
drcdths  of  it  ;  3  per  cent,  three-hundredths,  &c. 

256.  The  Rate  per  cent,  is  the  number  of  hundredths  taken. . 
Thus,  if    i    hundredth  is   taken,  the   rate   is  1  per  cent.  ;   if   2 
hundredths   are  taken,  the  rate  is  2  per  cent.;  if  3  hundredths, 
the   rate  is  3  per  cent.,  &c. 

257.  The  Base  is  the  number  whose  part  is  taken. 

258.  The  Percentage  is  the  result  of  the  operation,  and  is 
the  part  of  the  base  taken. 

The  rate  per  cent,  is  generally  expressed  decimally  ;  thus, 

i  per  cent,  of  a  number,  is  j-Jq  of  it  =  .01       of  it. 

2  per  cent,  of  a  number,  is  j^^  of  it  =  .02       of  it. 

25  per  cent,  of  a  number,  is  ^^^  of  it  =  .25       of  it. 

50  per  cent,  of  a  number,  is  -^q^j  of  it  =  .50       of  it. 

100  per  cent  of  a  number,  is  ^^J  of  it  =  1       time  it. 

200  per  cent  of  a  number,  is  f gj  of  it  =  2     times  it. 

-J  per  cent,  of  a  number,  is  jg^  of  it  =  .005     of  it. 

f  per  cent,  of  a  number,  is  j^q  of  it  =  .0015  of  it. 

.75  per  cent,  of  a  number,  is  ^  of  it  —  .0075  of  it. 

.S^  per  cent,  of  a  number,  is  j^^'^  of  it  =  .0085  of  it. 

Note. — Per  cent,  is  often  expressed  by  the  character  %.  Thus  5 
per  cent,  is  written  5  % ;  8  per  cent.,  8  %• 

Write,  decimally,  5%;  8%;   151%;   100%;  204%;  3271-% 
672.3 Vo 4  49%;  and  507.5%. 


255.  What  is  the  meaning  of  per  cent.?  What  is  5  hundredths  of 
a  number? — 256.  What  is  the  rate  per  cent.?  If  four  hundredtlis  of 
a  number  is  taken,  what  is  the  rate? — 257.  What  is  the  base? — 258. 
What  is  the  percentage?  How  is  the  rate  per  cent,  generally  ex 
pressed  ? 


FEKCKNTAGE. 


231 


259.   To  find  the  percentage,  when  the  base  and  rate  are  known. 

1.   What   is  the   percentage    of   $450,  the   rate  being  6  per 
cent.  ? 

Analysis. — The  rate,  expressed  decimally,  is  operation 
00.    The  percentage  is,  therefore,  six  hundredths  450 

of  the   base,    or  the  product  of   the  base   and  .06 

rate.      Hence,   to   find    the  percentage    of    any  ^27.00  Ans 
nmnber: 

Rule. — Mulfvply  the  base  by  the  rate,  and  the  product  will 
he  the  percentage. 

Examples. 


Find  the  percentage  of  the 

1.  4  per  cent,  of  §1256. 

2.  12%   of  $956.50. 

3.  I  per  cent,  of  475  yards. 


of  324.5  cwt. 


JVo   of  125.25  lbs. 
If  per  cent,  of  750 bush. 
4^Vo   of  $2000. 
9  per  cent,  of  186  miles. 
9.  lOJ  per  cent,  of  460  sheep. 

10.  5j^Q  per  cent,  of  540  tons. 

11.  82-  per  cent,  of  $3465.75. 


following  numbers  : 

12.  12^%  of  126  cows. 

13.  50  per  cent,  of  320  bales. 

14.  37^  per  cent,  of  1275  yds. 

15.  95  7o   of  $4573. 

16.  105  per  cent,  of  2500  bar. 

17.  112i%  of  $4573. 

18.  250  per  cent,  of  $5000. 

19.  305%   of  $1267.871 

20.  500  per  cent,  of  $3000. 

21.  What  is  3%  of  $765? 

22.  What  is  4^  %  of  960  bush.  ? 


•:;;  What  is  the  difference  between  4f "/«  of  $1000  and  7 J 
,..'r  cent,  of  $1500? 

24.  If  I  buy  895  gallons  of  molasses,  and  lose  17  per  cent. 
by  leakage,  how  much  have  I  left? 

25.  A  grocer  purchased  250  boxes  of  oranges,  and  found  (li;i' 
lie  had  lost  in  bad  ones  18  per  cent. :  how  many  full  boxes  d 
good  ones  had  he  left  ? 


259.  How  do  you  find  the  percentage  from  tho  base  and  raio? 


232  PERCENTAGE. 

260.    Parts  of  percentage. 
There    are    three   part?   in   percentage  :    1st.  The   Base ;  2d. 
The  Rate  ;  and  3d.  The  Percentage. 

26 1 .    To  find  the  rate,  when  the  base  and  percentage  are  known. 

I.  What  per  cent,  of  $64  is  116?   or,  $16  is  what  part  of 
$64? 

Analysis. — In   this    example,    16    is    the  opeuation. 

percentage,  and  64  the  base,  and  the  rate  is  1 6  —  i  —  25^  or 

required.     Since  the  percentage  is  equal  to  25  per  cent, 

the  base  multiplied  bj  the  rate,  the  rate  is 
equal  to  the  quotient  of  the  percentage  divided  by  the  base. 

Rule. — Divide  the  percentage  hy    the  base,   and  the  first 
two  decimal  places  ivill  express  the  rate. 

Examples. 

1.  What  per  cent,  of  10  dollars  is  2  dollars  ? 

2.  What  per  cent,  of  32  dollars  is  4  dollars  ? 

3.  What  per  cent,  of  40  pounds  is  3  pounds  ? 

4.  Seventeen  bushels  is  what  per  cent,  of  125  bushels  ? 

5.  Thirty-six  tons  is  what  per  cent,  of  144  tons  ? 

6.  What  per  cent,  is  $84  of  $96  ? 
T.  What  per  cent,  is  J  of  -I  ? 

8.  What  per  cent,  is  3  miles  of  400  miles  ? 

9.  Four  and  one-third  is  what  per  cent,  of  9J  ? 

10.   One  hundred  and   four    sheep   is   what    per    cent,    of   a 
drove  of  312  sheep? 

II.  A  grocer  has  $325,  and  purchases  sugar  to  the  amount 
of  $12 1.87 J  :   what  per  cent,  of  his  money  does  he  expend? 

12.    Out  of  a  bin  containing  450  bushels  of  oats,  56 J  bushels 
were  sold  :    what  per  cent,  is  this  of  the  whole  ? 

Note. — If  the  base  be  regarded  as   a  single  tiling,  and  denoted  by 
1,  a  fractional  percentage  expressed  decimally  will  denote  the  rate. 


2G0.   How  many  parts  are  there  in  percentage?    What  are  tliey? 
2C1.  How  do  you  find  the  rate,  ^om  the  base  and  percentage? 


PERCENTAGE.  233 

13.  J  of  a  number  is  what  per  cent,  of  the  number  ? 

14.  J  of  a  sliip  is  what  per  cent,  of  the  ship  ? 

15.  ^Q  of  50  is  what  per  cent,  of  50? 

16.  f  of  a  cargo  is  what  per  cent,  of  it? 

n.    1|  times  a  number  is  what  per  cent,  of  the  number? 

262.  To  find  the  base,  when  the  rate  and  percentage  are 
known. 

1.  Of  what   number  is  $960,  16  per  cent.? 
Analysis. — By  Art.  259,  the  percentage 

,  ,  ,.,.-,,  r  OPERATION. 

IS   equal    to  tlie  base    multiplied  by  the        -,^-,         ,_        »„„_ 
rate ;   hence,  to  find  the  base, 

Rule. — Divide  the  percentage  hy  the  rate,  expressed  deci- 
molly. 

Examples. 

2.  The  number  475  is  25%   of  what  number? 

3.  The  number  87J  is  12^ 7o   of  what  number? 

4.  Five  hundred  and  sixty  dollars  is  140%  of  what  number? 

5.  The  number  75  is  J°/o  of  what  number? 

6.  One  dollar  and  twenty-five  cents  is  -J  %  of  what  number  ? 

7.  The   fraction   |    is   45%    of  what   number? 

8.  The  fraction  }  is  f %   of  what  number? 

9  If  a  person  receives  $5850,  and  that  sum  is  75  Vo  of 
what  is  due  him,  what  is  the  debt  ? 

10.  A  bankrupt  can  pay  only  37^  per  cent,  of  his  debts  : 
what  did  he  owe  to  that  merchant  to  whom  he  paid  $1647  ? 

11.  In  an  army,  15600  men  are  mustered  after  a  battle,  in 
Avhich  25  %  were  killed  and  wounded  :  what  was  the  original 
number  of  men  ? 

263.  Amount  is  the  percentage  plus  the  base. 

264.  Difference  is  the  percentage  minus  the  base. 

262.  How  do  you  find  the  base,  when  the  rate  and  percentage  are 
known  ?— 26a.   What  is  the  amount?— 264.   ^^^lat  is  tlie  difierence? 


230  rEKCENTAGE. 

2.  Mr.  Wilson  lost  18  per  cent,  of  Lis  sheep  by  disease,  and 
had  a  flock  of  615  left  :    how  many  had  he  at  first  ? 

3.  Mr.  Jones  invests  46''/o  of  his  capital  in  land,  and  has 
$0 1 3  left  :    what  is  his  capital  ? 

4.  An  army  fought  two  battles  j  in  the  first  it  lost  15  per 
cent.,  and  in  the  second,  20  per  cent,  of  the  original  number  ; 

fter    which    it   mustered    19500   men  :    what   was   its  original 
strength  ? 

5.  A  grocer  bought  a  quantity  of  provisions,  but  finding 
them  damaged,  sold  them  at  a  loss  of  19  per  cent.,  and  re- 
ceived $10935  :   what  did  they  cost  him  ? 

6.  A  son,  who  inherited  a  fortune,  spent  37 1  per  cent,  of 
it,  when  he  found  that  he  had  only  131250  remaining  :  what 
was  the  amount  of  his  fortune  ? 

t.  A  grocer  purchased  a  lot  of  teas  and  sugar,  on  which 
he  lost  16  per  cent,  by  selling  them  for  $4200  :  what  did  he 
pay  for  the  goods  ? 

8.  A  speculator  invested  in  stocks,  which,  falling  rapidly  in 
price,  he  sold  out  at  a  loss  of  13  per  cent.,  and  received 
$2262  :    what  was  the  amount  of  his  purchase  ? 

267.     Formulas  of  percentage. 

Nearly  every  practical  question,  in  Arithmetic,  is  a  partic- 
ular case  of  one  or  other  of  the  five  operations  of  Percentage : 
hence,  we  write  the  formulas  : 

1.  Percentage  =  Base  X  Rate Art.  259. 

2.  Rate  =  Z^'ES!  Art.  261. 

Base 

o    -r.  Percentaore  ,   ,    ^^^ 

3.  Base  =  — ^  ^    °  Art.  262. 

Kate 

A     -o  Amount  .   ..    o/»r 

4    Base  = =r— -  Art.  265. 

1  4-  Bate 

e     -o  Difference  .    ,     ^^^ 

6.   Base  =  :. =i— -  -         -    -    -    Art.  266. 

1  —  Kate 


PKOFIT  AND   LOSS.  237 


PROFIT    AND     LOSS. 

268.  Profit  and  Loss  are  commercial  terms,  indicating  gain 
or  loss  in  busiiicss  transactions.  The  gain  or  loss  is  always 
estimated  on  the  cost  price. 

The  cost  of  an  article  is  the  amount  paid  for  it. 

The  selling  price  of  an  article  is  the  amount  received  for  it. 

Tlie  cost  is  the  base ;  the  gain  or  loss  is  the  percentage  ; 
the  rate  per  cent,  of  gain  or  loss  is  the  rate;  the  selUng  price 
is  the  sum  of  the  base  and  percentage,  when  there  is  a  gain, 
and  their  difference  when  there  is  a  loss. 

The  following  examples  may  all  be  wrought  by  the  five  for- 
umlas  and  rules  of  Percentage  : 

Examples. 

1.  Bought  9  barrels  of  sugar,  each  weighing  250  pounds,  at 
7  cents  a  pound  :  how  much  profit  would  be  made  if  it  were 
sold  at  8 1  cents  per  pound? 

2.  If  15  pieces  of  muslin,  each  containing  43  yards,  cost  2t 
cents  per  yard,  what  would  be  the  gain  if  sold  at  31i  cents 
per  yard? 

3.  A  farmer  bought  a  flock  of  360  sheep  ;  their  keeping 
for  1  year  cost  $0.75  a  head  ;  their  wool  was  worth  1  dollar 
and  25  cents  a  head,  and  one-fourth  of  them  had  lambs,  each 
of  which  was  worth  one-half  as  much  as  a  fleece  :  what  was 
the  profit  of  the  purchase  at  the  end  of  the  year? 

4.  A  merchant  bought  65  barrels  of  flour,  at  $5J  per  barrel, 
and  sold  them  so  as  to  gain  $42.50  :  what  was  the  price  per 
barrel  ? 

5:    A  person  bought  500    bushels   of   potatoes,  at  62^  cents 


2G8  What  do  you  understand  by  the  terras  profit  and  loss? 
What  is  cost?  What  is  the  selling  price?  What  is  the  base?  What 
is  the  gain  or  loss? 


238  PERCENTAGE. 

per  bushel,  and  sold  them  so  as  to  gain  $35  :  at  what  price 
were  they  sold? 

6.  A  house  and  lot  were  bought  for  $6450.  The  house 
was  repaired  at  an  expense  of  $5t5,  painted  for  $796,  and 
was  then  sold  so  as  to  gain  $945  :  what  was  the  price  of  the 
house  and  lot? 

T.  If  in  3  hogsheads  of  molasses,  which  cost  $68.04,  ono 
third  leaked  out,  what  must  the  remainder  be  sold  for  per 
gallon  to  realize  a  profit  of  $2.52  on  the  whole? 

8.  If  a  merchant's  profits  are  22  per  cent,  on  the  cost  of 
the  goods  sold,  what  is  his  profit  on  $4162.50  ? 

9.  A  quantity  of  goods  were  bought  for  $3612  ;  the  charges 
on  them  were  $54  :  they  were  sold  at  an  advance  of  20  %  ; 
what  was  the  profit? 

10.  A  merchant,  on  taking  an  inventory  of  stock,  finds  it 
worth  $37649 :  what  would  be  his  profit  on  this  stock,  if 
sold  at  an  advance  of  31|-Vo? 

11.  A  merchant  bought  goods  to  the  amount  of  $2965  ;  but 
being  damaged,  he  sold  them  at  a  loss  of  15%  :  what  was  the 
amount  of  his  loss? 

12.  A  quantity  of  flour  was  bought  for  $8550  ;  J^  of  the 
flour  was  so  damaged  as  to  be  sold  at  a  loss  of  12"/o ;  -^  of  it 
was  sold  at  a  profit  of  19%;  and  the  remainder  at  a  profit 
of  30  %  :   what  was  the  net  profit  on  the  flour  ? 

13.  If  sugar  costs  10  cents  per  pound,  and  is  sold  at  an 
advance  of  12^%,  what  is  the  profit  per  lb.? 

14.  A  bank,  whose  capital  is  $200000,  after  reserving  $2860 
for  a  surplus  fund,  declared  a  dividend  of  8  %  on  the  capital : 
what  were  the  entire  profits  ? 

15.  The  profits  of  a  merchant  averaged  25%  of  his  capital  ; 
and  his  expenses  are  5  %  of  his  profits  :  what  part  of  tho 
capital  were  the  expenses? 

16.  A  farmer  sells  375  bushels  of  corn  for  75  cents  a  bushel  j 


PliOFlT    AND    LOSS.  23& 

the  piircliaser  sells  it  at  au  advance  of  20  per  cent.:  how  much 
a  bu^^iiel  did  he  receive  for  the  corn? 

n.  A  merchant  buys  a  pipe  of  wine,  for  which  he  pays 
$322.56,  and  he  wishes  to  sell  it  at  an  advance  of  25  per  cent.  : 
what  must  he  sell  it  for  per  gallon  ? 

18.  A  man  bought  3215  bushels  of  wheat,  for  which  he  paid 
$3493.33J,  but  finding  it  damaged,  is  willing  to  lose  10  per 
cent. :  what  must  he  sell  it  for  per  bushel  ? 

19.  If  I  purchase  two  lots  of  laud  for  $150.25  each,  and  sell 
one  for  40  per  cent,  more  than  it  cost,  and  the  other  for  28 
per  cent,  less,  what  is  the  gain  on  the  two  lots  ? 

20.  Bought  a  cask  of  molasses  containing  144  gallons,  at  45 
cents  a  gallon,  36  gallons  of  which  leaked  out :  at  what  price 
per  gallon  must  I  sell  the  remainder  to  gain  10  per  cent,  on 
the  cost  ? 

21.  A  person  in  Chicago  bought  3500  bushels  of  wheat,  at 
$1.20  a  bushel:  allowing  5  per  cent,  on  the  cost  for  risk  in 
transportation,  3  per  cent,  for  freight,  and  2  per  cent,  commis- 
sion for  selling,  what  must  it  be  sold  for  per  bushel  in  New 
York  to  realize  40  per  cent,  net  profit  on  the  purchase  ? 

22.  Bought  a  quantity  of  goods  for  1348.50,  and  sold  the 
same  for  8425  :  what  per  cent,  did  I  make  on  the  amount 
received  ? 

23.  Bought  a  piece  of  cotton  goods  for  6  cents  a  yard,  and 
sold  it  for  7 J  cents  a  yard  :  what  was  my  gain  per  cent.? 

24.  If  I  buy  rye  for  90  cents  a  bushel,  and  sell  it  for  $1.20, 
and  wheat  for  $1.1 2^  a  bushel,  and  sell  it  for  $1.50  a  bushel, 
uj)on  which  do  I  make  the  most  per  cent.? 

25.  If  paper  that  cost  $2  a  ream,  be  sold  for  18  cents  a 
quire,  what  is  gained  per  cent.  ? 

26.  How  much  per  cent,  would  be  made  upon  a  hogshead  of 
sugar  weighing  13cwt.  3  qr.  141b.,  that  cost  $8  per  cwt.,  if 
sold  at  10  cents  per  pound  ? 

27.  A  hardware  merchant  bought  45  T.  16  cwt.  25  lb.  of  iron. 


2tl:0  PERCENTAGE. 

at  $75  per  ton,  and  sold  it  for  $78.50  per  ton  :  what  was  his 
whole  gain,  and  how  much  per  cent,  did  he  make  ? 

28.  A  merchant  buys  67560  feet  of  lumber  for  $7000  :  the 
expense  of  cartage  and  piling  was  |425,  and  the  loss  of  material 
amounted  to  $216.  If  the  lumber  be  sold  at  $97.50  per  1000 
feet,  Avhat  will  be  the  entire  loss  ? 

29.  A  gentleman,  having  gold  coin  to  the  amount  of  $475, 
sold  it  for  bank  bills  and  obtained  $593.75  :  what  was  the  rate 
per  cent,  of  premium  on  gold,  and  what  the  rate  per  cent,  of 
depreciation  on  the  bills  ? 

30.  In  selHng  a  quantity  of  wheat,  a  merchant  gained  $500 
when  his  rate  of  profit  was  31%:  what  was  the  cost? 

31.  In  the  course  of  6  months  a  merchant  gained  $3745  : 
what  amount  of  goods  must  he  have  sold,  allowing   a  gain  of 

25  Vo? 

32.  The  net  profits  of  a  shoe-dealer  were  $2965,  and  his  ex 
penses  were  $1260.  If  the  rate  of  profit  were  40%,  what 
amount  of  goods  were  sold  ? 

33.  What  must  be  the  annual  sales  of  a  merchant,  that  he 
may  realize  $4500,  after  paying  $2500  expenses,  when  his  rate 
of  profit  is  35  %  ? 

34.  The  surplus  fund  of  an  insurance  company,  amounting  tc 
$32500,  will  pay  12^%   on  its  capital:  what  is  the  capital? 

35.  The  profits  of  a  bank  are  12%  of  its  capital  ;  the  ex- 
penses are  10%  of  the  profits  :  what  Vo  of  the  capital  are  the 
expenses  ? 

36.  A  grocer  sold  a  lot  of  sugars  for  $477.12,  which  was  aa 
advance  of  12%  on  the  cost:  what  was  the  cost? 

37.  Mr.  A.  bought  a  lot  of  sugars,  but  finding  them  of  an 
hiferior  qaulity,  sold  them  at  a  loss  of  15%,  and  found  that 
they  brought  $340  :    what  did  they  cost  him  ? 

38.  I  sold  a  parcel  of  goods  for  $195.50,  on  which  I  made 
15%:   what  did  they  cost  me? 


PROFIT   AND   LOSS.  241 

39.  Sold  18  cwt,  3  qr.  14  lb.  of  sugar,  at  8  cents  a  pound, 
and  gained  15%:    how   much  did  the  whole  cost? 

40.  A  dealer  sold  two  horses  for  $412.50  each,  and  gained 
on  one  35 '/o,  but  lost  10 Vo  on  the  other:  what  was  the  cost 
of  each,  and  what  was  his  net  gain  ? 

41.  A  merchant  havmg  a  lot  of  flour,  asked  dS^Vo  more  than 
it  cost  him,  but  was  obliged  to  sell  it  12^%  less  than  hi3 
asking  price  :  he  received  $1  per  bbl. :  what  was  the  cost 
per  bbl.? 

42.  If  a  merchant  in  selling  a  quantity  of  merchandise  for 
$3850,  loses  12%  of  the  cost,   what  was  the  cost? 

43.  If  25  per  cent,  be  gained  on  flour  when  sold  at  $10  a 
barrel,  what  per  cent,  would  be  gained  when  sold  at  $11.60  a 
barrel  ? 

Note. — In  this  class  of  examples,  first  find  the  cost,  as  in  Art.  267 ; 
then  find  the  gain,  or  losg;  and  tlien  divide  bj  the  number  on  which 
the  per  cent,  is  reckoned. 

44.  A  lumber-dealer  sold  25650  feet  of  lumber  at  $19.20  a 
thousand,  and  gained  20  per  cent.  :  how  much  would  he  have 
gained  or  lost  had  he  sold  it  at  $15  a  thousand? 

45.  A  man  sold  his  farm  for  $3881.25,  by  which  he  gamed 
12J  per  cent,  on  its  cost :  what  was  its  cost,  and  what  would 
he  have  gained  or  lost  per  cent,  if  he  had  sold  it  for  $3211.60? 

46.  If  a  merchant  sells  tea  at  66  cents  a  pound,  and  gains  20 
per  cent.,  how  much  would  he  gain  per  cent,  if  he  sold  it  at 
^1  cents  a  pound  ? 

41.  Sold  5520  bushels  of  corn  at  50  cents  a  bushel,  and 
lost  8  per  cent.:  how  much  per  cent,  would  have  been  gained 
had  it  been  sold  at  60  cents  a  bushel  ? 

48.  A  grocer  bought  3  hogsheads  of  sugar,  each  weighing 
11121  pounds  ;  he  sold  it  at  11  cents  a  pound,  and  gained  31^ 
per  cent.  :  what  was  its  cost,  and  for  how  much  should  he 
have  sold  it  to  gain  50  per  cent,  on  the  cost  ? 

11 


242  PERCENTAGE. 


COMMISSION. 

269.  Commission  is  an  allowance  made  to  an  agent  for  a 
transaction  in  business,  and  is  reckoned  at  a  certain  rate  per 
cent,  on  tlie  amount  of  money  used. 

270.  A  Commission  Merchant  is  one  who  sells  or  buys 
goods  for  another. 

271.  A  consignment  is  a  quantity  of  goods  sent  to  a 
merchant  for  sale. 

A  consignor  is  the  one   who  sends  the  goods. 

A  consignee  is  the  one  to  whom  the  goods  are  sent. 

Note. — The  commission  for  the  purchase  or  sale  of  goods,  in  the 
city  of  New  York,  varies  from  2h  to  12^  per  cent. ;  and,  under  some 
circumstances,  even  higher  rates  are  paid.  For  the  sale  of  real  estate 
the  rates  are  lower,  varying  from  one-quarter  to  2  per  cent. 

All  the  cases  of  Commission  come  under  the  rules  and  for- 
mulas of  Percentage. 

1.  A  commission  merchant  sold  a  lot  of  goods,  for  which 
he  received  $1540  ;  he  charged  2^  per  cent,  commission  :  what 
was  the  amount  of  his  commission,  and  how  much  must  he 
pay  over? 

2.  A  commission  merchant  receives  $1399.11  to  be  invested 
in  groceries  ;  he  is  to  receive  3  per  cent,  on  the  amount  of  the 
purchase :  what  amount  is  laid  out  in  groceries,  and  what  the 
commission  ? 

3.  An  auctioneer  sold  a  house  for  $3125,  and  the  furniture 
for  $1520  :   what  was  his  commission  at  f  per  cent.  ? 

4.  A  flour  merchant  sold  on  commission  150  barrels  of 
flour,  at  $9.15  a  barrel  :  what  was  his  commission  at  2 J-  per 
cent.  ? 

269.  Wliat  is  commission  ?  How  is  it  reckoned  ? — 270.  What  is  a 
commission  merchant? — 271.  What  is  a  consignment?  What  is  a  con- 
Bignor  ?    ^^^lat  is  a  consignee  ? 


COMMISSION.  243 

6.  I  sold  at  auction  96  hogsheads  of  sugar,  each  weighing 
9  cwt.  and  50  lb.,  at  $6.50  per  hundred :  what  was  the 
auctioneer's  commission  at  If/o,  and  to  how  much  was  I 
entitled  ? 

6.  An  agent  purchased  2340  bushels  of  wheat  at  $1.75  a 
bushel,  and  charged  2-|  per  cent,  for  buying,  IJ  per  cent,  fof 
shipping,  and  the  freiglit  cost  2  per  cent. :  what  was  his  com 
mission,  and  what  did  the  wheat  cost  the  owner  ? 

t.  A  town  collector  received  4 J  per  cent,  for  collecting  a 
tax  of  $2564.25  :    what  was  the  amount  of  his  percentage  ? 

8.  I  paid  an  attorney  6f  per  cent,  for  collecting  a  debt  of 
$7320.25  :    how  much  did  I  receive  ? 

9.  ^ly  commission  merchant  sold  goods  to  the  amount  of 
.4000,  on  which  I  allowed  him  5  per  cent. ;  but  as  he  paid  over 
the  money  before  it  became  due,  I  allowed  him  1 J  per  cent, 
more  ;  how  much  am  I  to  receive  ? 

10.  A  dairyman  sent  an  agent  3476  pounds  of  cheese,  and 
allowed  him  3J  per  cent,  for  selling  it :  how  much  would  he 
receive  after  deducting  the  commission,  if  it  were  sold  for  12 J 
cents  per  pound  ? 

11.  A  person  has  $1500  in  bills  of  the  State  Bank  of  In- 
diana, upon  which  there  is  a  discount  of  2J  per  cent.,  and 
$1000  of  the  bank  of  Maryland,  upon  which  there  is  a  discount 
of  3J-  per  cent.  :  what  will  be  the  loss  in  changing  the  amount 
into  current  money? 

12.  I  am  obliged  to  sell  $2640  in  bills  on  the  bank  of  Dela- 
ware, upon  which  there  is  a  discount  of  2f  per  cent.  :  how 
much  bankable  money  should  I  receive  ? 

13.  A   merchant   in   New  York  received  a    consignment   Oi 
75  bbl.  of  flour,  which  he  sold  at  $4.75  per  bbl.     He  charged 

I  commission  of  2%,  1%  for  storage,  and  f'/o  for  guarantee 
\Vhat  were  the  charges,  and  what  amount  was  transmitted  to 
ibe  consignor? 

Note, — 2  per  cent,  -f  i  per  cent.  +  }  per  cent.  =  8  per  cent 


244  PERCENTAGE. 

14.  A  commission  merchant  in  'Naw  York  receives  $12000 
for  tlie  purchase  of  sugar.  He  charges  2%  commission.  What 
amount  is  laid  out  in  purchasing  sugar,  and  what  is  the  com- 
mission ? 

15.  A  factor  receives  1108.75,  and  is  directed  to  purchase 
iron  at  $45  a  ton  ;  he  is  to  receive  5  per  cent,  on  the  money 
paid  :  how  much  iron  can  he  purchase  ? 

16.  I  forward  $2608.625  to  a  commission  merchant  in  Chicago, 
requesting  him  to  purchase  a  quantity  of  corn  ;  he  is  to  receive 
2^  per  cent,  on  the  purchase  :  what  does  his  commission  amount 
to,  and  how  much  corn  can  he  buy  with  the  remainder,  at  56 
cents  a  bushel? 

n.  My  agent  at  Havana  purchased  for  me  a  quantity  of 
sugar  at  6^  cents  a  pound,  for  which  I  allow  him  a  com- 
mission of  IJ  per  cent.  His  commission  amounts  to  $42.66  : 
how  many  barrels  of  sugar  of  240  pounds  each  did  he  purchase, 
and  how  much  money  must  I  send  him  to  pay  for  it,  including 
his  commission? 

18.  A  merchant  in  New  Orleans  received  $187.50,  to  be  laid 
out  in  the  purchase  of  cotton.  After  allowing  for  commission 
at  2Vo,  freight  at  i%,  insurance  at  J%,  and  incidental  expenses 
■j^qVo,  what  amount  was  expended  in  the  purchase  of  cotton, 
and  what  was  the  commission? 

19.  A  commission  merchant,  in  selling  a  quantity  of  mer- 
chandise for  $2*185,  received  a  commission  of  $60  :  what  was 
the  rate  of  commission  ? 

Note. — In  this  example,  the  base  and  percentage  are  given,  and 
the  rate  is  required  (Art.  267). 

20.  A  land  agent  received  $175  for  selling  a  house  for 
$6795  :  what  was  his  rate  of  commission? 

21.  A  collecting  agent  received  $15  for  collecting  a  debt  of 
$175  :  what  was  his  rate  of  commission? 

22.  A  miller  received  for  his  toll  5  bushels  on  every  45 
bushels  of  grain  that  he  ground  :  what  was  the  rate  ? 


i 


INTEREST.  245 


INTEREST. 

272.  Interest  is  a  percentage  paid  for  the  use  of  money 
Principal,  or  base,  is  the  money  on  which  interest  is  paid. 
Rate  of  interest  is  the  per  cent,  paid  per  year. 

Amount  is  tlie  sum  of  the  principal  and  interest. 
Per  annum  means  by  the  year. 

273.  In  interest,  by  general  custom,  a  year  is  reckoned  at  12 
niunths,  each  having  30  days.  The  Rate  of  Interest  is  generally 
lixed  by  law,  and  is  called  Legal  Interest.  Any  rate  above 
the  legal  rate  is  usury,  and  is  generally  forbidden  by  law. 

274.  In  most  of  the  States,  the  legal  rate  is  6  per  cent.  ;  in 
New  York,  South  Carolina  and  Georgia,  it  is  7  % ;  and  in 
some  of  the  other  States  the  rate  is  fixed  as  high  as  10  per 
cent. 

275.  There  are  five  parts  in  interest :  1st,  principal ;  2d,  rate ; 
3d,  time  ;   4th,  interest  ;    5th,  amount. 

CASE    I. 

276.  To  find  the  interest  of  any  principal  for  one  or  more 
years 

1.  What  is  the  interest  of  13920  for  2  years,  at  7  per 
cent.  ? 

Analysis. — In    tliis    example,   the  operation. 

base  is  $3920,  the  rate  7%,  and  the         |3920 
interest  for  1  year  is  the  percentage :  q>j  xqXq, 

this  product  multiplied  by  2,  the  num- . 

!,er  of  years,  gives  the  interest  for  2      ^^74.40  mt.  for  1  year. 

years;    hence,  ?  ^^'  ^^  ^^ars. 

$548.80  interest. 

Rule. — Midtiply  the  principal 
by  the  rate,  expressed  decimally,  and  the  product  by  the  numr 
ber  of  years. 


246  PERCENTAGE. 

Examples. 

1.  What  is  the  interest  of  $6t5  for  1  year,  at  6|  per  cent.? 

2.  What  is  the  interest  of  |8tl.25,  for  1  year,  at  r/o? 
3    What  is  the  interest  of  1535.50,  for  7  years,  at  6%? 

I  What  is  the  interest  of  $1125.885,  for   4  years,  at  8%? 

6.  What  is  the  interest  of  $789.U,  for   12  years,   at    5% 

6.  What   is   the   interest   of  $2500,   for   7   years,  at   7|Vo? 

7.  What  is  the  interest  of   $3153.82,  for  2  years,  at  4i%? 

8.  What  is  the  amount  of  $199.48,  for  16  years,  at  7%? 

9.  What  is  the  amount  of  $897.50,  for  3  years,  at  8%? 

10.  What  is  the  interest  of  $982.35,  for  4  years,  at  6J%? 

11.  What  is  the  amount   of  $1500,    for   5   years,  at    5i%? 

12.  Wiiat  is  the  interest  of  $1914.10,  for  6  years,  at  ^'/o? 

13.  What  is  the  interest  of  $350,  for  21  years,  at  10%? 

14.  What  is  the  amount  of  $628.50,  for  5  years,  at  12i%? 

15.  What  is  the  amount  of  $75.50,  for  10  years,  at  6%? 

16.  What  is  the  amount  of  $5040,  for  2  years,  at  7|%? 

Note. — Wlien  there  are  years  and  months,  and  the  months  are 
an  aliquot  part  of  a  year,  multiply  the  interest  for  1  year  hy  t7i4 
years  and  the  montJis,  reduced  to  the  fraxXion  of  a  year. 

17.  What  is  the  interest  of  $119.48,  for  2yrs.  6  mo.,  at  7Vo? 

18.  What  is  the  interest  of  $250.60,  for  1  yr.  9  mo.,  at  6%? 

19.  What  is  the   interest  of  $956,  for  5yrs.  4  mo.,  at  9%? 

20.  What  is  the  amount  of  $1575.20,  for  3yrs.  8  mo.,  at  7%? 

21.  What  is  the  amount  of  $5000,  for'2yrs.  3  mo.,  at  5J%? 

22.  What  is  the  interest  of  $1508.20,  for  4  yrs.  2  mo.,  at  10%? 

23.  What  is  the  interest  of  $75,  for  6  yrs.  10  mo.,  at  12i% 

24.  What  is  the  amount  of  $125,  for  5  yrs.  6  mo.,  at  4f%? 

272.  What  is  interest?  What  is  the  principal  or  base?  What  is 
rate?  What  is  amount?  What  is  the  meaning  of  per  annum? — 273. 
How  is  a  year  reckcmed,  in  computing  interest?  How  many  days  are 
reckoned  in  a  month  ?    What  is  legal  interest  ?    What  is  usury  ? 


INTEREST.  247 

CASE     II. 

277.  To  find  the  interest  on  a  given  principal  for  any  rate 
and  time. 

1.  What  is  the  iuterest  of  $1752.95,  at  6  per  cent.,  for 
2yrs.  4  mo.  and  29da.  ? 

Analysis. — The  interest  for  1  year  is  the  product  of  the  prin 
oipal  and  rate.  If  the  interest  for  1  year  be  divided  by  12,  the 
quotient  will  be  the  interest  for  1  month ;  if  the  interest  for  1  month 
be  divided  by  30,  the  quotient  will  be  the  interest  for  1  day. 

The  interest  for  2  years  is  two  times  the  interest  for  1  year ;  the 
Interest  for  4  months,  4.  times  the  interest  for  1  month;  and  the 
interest  for  29  days,  29  times  the  Interest  for  1  day. 

OPERATION. 

$1752.96 
.06 


12)105,1776  int.  for  lyr.    $105.1776    x2    =$210.3552    2a/r. 
30)877648  int.  for  Imo.          8.7648    X  4    =      35.0592    47no. 
.29216  int.  for  Ida.  0.29216  X  29  =       8.47264  29da, 

Total  interest,     $253.88704 
Hence,  we  have  the  following, 

Rule. 

I.  Find  the  interest  for  1  year: 

II.  Divide  this  interest  by  12,  and  the  quotient  will  be  the 
interest  for  1  month: 

III.  Divide  the  interest  for  1  month  by  30,  and  the  quo- 
tient will  be  the  interest  for  1  day : 

IV.  Multiply  the  interest  for  1  year  by  the  member  of 
years,  the  interest  for  1  month  by  the  numbei'  of  months,  and 
the  interest  for  1  day  by  the  number  of  days,  and  the  sum 
f  the  products  will  be  the  required  interest. 

274.  What  are  the  general  rates  of  legal  interest  ? — 275.  How  many 
parts  arc  there  in  interest?  What  are  they? — 276.  How  do  you  find 
the  interest  of  any  principal  for  one  or  more  years? — ^77.  How  do  you 
find  the  interest  for  any  rate  and  time? 


248  PERGKKTAGE. 

Note. — Tliis  method  of  computing  interest  for  days,  is  the  one  \»» 
general  use.  It  supposes  the  month  to  contain  30  days,  or  the  year 
360  days;  whereas,  it  actually  contains  865  days. 

To  find  the  exact  interest  for  1  day,  we  must  regard  the  month  as 
containing  ^2^  days  =  30j^2  ^^J^  5  ^^^  *^s  is  *^®  number  by  which 
the  interest  for  one  month  should  be  divided,  in  order  to  find  the 
''.met  interest  for  one  day.  As  the  divisor,  commonly  used,  is  too  small, 
lie  interest  found  for  1  day,  is  a  trifle  too  large.  If  entire  accuracy 
required,  the  interest  for  the  days  must  be  diminished  by  its 
jIs  part  =  ^3  part. 

2d  method. 

278.  There  is  another  rule  resulting  from  liie  last  analysis 
which  is  regarded  as  the  best  general  method  of  computing  interest. 

Rule. 

I.  Find  the  interest  for  1  year,  and  divide  it  &?/  12  ;  the 
quotient  will  he  the  interest  for  1  month: 

II.  Multiply  the  interest  for  1  month  by  the  time  expressed 
in  months  and  decimal  parts  of  a  month,  and  the  product 
will  be  the  required  interest. 

Note. — Since  a  month  is  reckoned  at  80  days,  any  number  of  days 
is  redticed  to  decim<iU  of  a  month  by  dividing  the  number  of  days  by  3. 


Examples. 
1,  Whftt  is  the  interest  of  $655,  for  3  years  1  months  and 
3  days,  at  1%2 

OPERATION. 

3  ?/?'s.  =  36  mos.                          1655 
t  mos.                             .01 

3  da.   =  .4|-^2 OS. 

12)45.85            int.  for  1  year. 

Time  =  43.4J  mos. 

3.82083+  int.  for  1  month. 
43.4i    time  in  months. 

127361 
1528332 
1146249 
1528332 

165.951383  Ans. 

INTEREST.  249 

2.  What  is  the  interest  of  $358.50,  for  1  yr.  8  mo.  6da.,  at  T  %? 

3.  What  is  the  interest  of  $1461.75,  for  4  yrs.  9  mo.  15  da., 
at  6  per  cent.? 

4.  What  is  the  interest  of  $1200,  for  2  years  4  months  and  12 
days,  at  IJVo? 

5.  What  is  the  interest  of  $4500,  for  9  mos.  20  da.,  at  5%? 

6.  What  is  the  interest  of  $156.25,  for  10  mo.  18 da.,  at  8%? 

7.  What  is  the  interest  of  $640,  for  3  yrs.  2  mo.  9  da.,  at  OJVo? 

8.  Wliat  is  the  interest  of  $276.50,  for  11  mo.  21  da.,  at  10%? 

9.  What  is  the  amount  of  $378.42,  for  1  yr.  5  mo.  3  da.,  at  7%  ? 

10.  What  is  the  amount  of  $1250,  for  7  mo.  21  da.,  at  10^%? 

11.  Wliat  is  the  interest  of  $6500,  for  2  mo.  10  da.,  at  91%  ? 

12.  What  is  the  interest   of   $70.50,  for    10    years    and    10 
months,  at  5^  per  cent.  ? 

13.  What  is  the  amount  of  $45,  for  12  years  and  27  days, 
at  Gj  per  cent.  ? 

14.  What  will   $100   amount  to  in    15  years  and  6  months, 
if  put  at  interest  at  4  per  cent.  ? 

15.  How  much  will  $475.50   gain   in   5  years  9  months  and 
24  days,  at  8  per  cent.  ? 

16.  What  will  be  the  interest  of  $4560,  for  14  months  and 
19  days,  at  7  per  cent.? 

17.  What  will  $128.37J   amount    to   in    10   months   and  27 
days,  at  6  per  cent.? 

18.  What  is  the  interest  of  $264.52,  for   2  years  8  months 
uud  14  days,  at  6  per  cent.? 

19.  What  is  the   amount   of  $76.50,  for   1   year   9   montlis 
ind  12  days,  at  6  per  cent.  ? 

20.  What  will  be  the  interest  for  3  years  3  months  and  15 
(lays,  of  $241.60,  at  7  per  cent.? 

21.  What  is   the   mterest  of  $5600,  for   30    days,  at  7  pc 
ut.  ? 

278.  What  is  the  rule  for  the  second  method? 


248  PEIiCKIS'TAGE. 

Note. — Tliis  method  of  computing  interest  for  days,  is  the  one  \m 
general  use.  It  supposes  the  month  to  contain  30  days,  or  the  year 
360  days;  whereas,  it  actually  contains  865  days. 

To  find  the  exact  interest  for  1  day,  we  must  regard  the  month  as 
containing  ^2^  days  =  80  ^^  ^^^®  5  ^^^  *^^  ^^  *^®  number  by  which 
the  interest  for  one  month  should  be  divided,  in  order  to  find  the 
^xact  interest  for  one  day.  As  the  divisor,  commonly  used,  is  too  small, 
lie  interest  found  for  1  day,  is  a  trifle  too  large.    If  entire  accuracy 

required,  the  interest  for  the  days  must  be  diminished  by  its 
56  3  part  =  ^ij  part. 

2d  method. 

278.  There  is  another  rule  ipesulting  from  ihe  last  analysis 
which  is  regarded  as  the  best  gen«ral  method  of  computing  interest. 

Rule. 

I.  Find  the  inter ed  for  1  year,  and  divide  it  by  12  -j  the 
quotient  will  be  the  interest  for  1  month: 

II.  Multiply  the  interest  for  1  month  by  the  time  expressed 
in  months  and  decimal  parts  of  a  month,  and  the  product 
will  be  the  required  interest. 

Note. — Since  a  month  is  reckoned  at  30  days,  an^  number  of  days 
is  reduced  to  decimals  of  a  month  ly  dividing  the  number  of  days  by  3. 

Examples. 
1,  Whftt  is  the  interest  of  |655,  for  3  years  7  months  and 
13  days,  at  r/o? 

OPERATION. 


3  yrs.  =  36  mos. 
1  mos. 

$655 
.01 

]^da.   =  A^mos. 

12)45.85            int.  for  1  year. 

Time  =  4 3.4  J  mos. 

3.82083+  int.  for  1  month. 
43.4i    tune  in  months. 

121361 
1528332 
1146249 
1528332 

165.951383  Ans. 

INTEREST.  249 

2.  What  is  the  interest  of  $358.50,  for  1  yr.  8  mo.  6da.,  at  1  %? 

3.  What  is  the  interest  of  $1461.75,  for  4  yrs.  9  mo.  15  da., 
at  6  per  cent.? 

4.  What  is  the  interest  of  $1200,  for  2  years  4  months  and  12 
days,  at  T^Yo? 

5.  What  is  the  interest  of  $4500,  for  9  mos.  20  da.,  at  5Vo? 

6.  What  is  the  interest  of  $156.25,  for  10  mo.  18 da.,  at  8%? 

7.  What  is  the  interest  of  $640,  for  3  yrs.  2  mo.  9  da.,  at  6J%? 

8.  What  is  the  interest  of  $276.50,  for  11  mo.  21  da.,  at  10%? 

9.  What  is  the  amount  of  $378.42,  for  1  yr.  5  mo.  3  da.,  at  7  %  ? 

10.  What  is  the  amount  of  $1250,  for  7  mo.  21  da.,  at  lO^Vc? 

11.  What  is  the  interest  of  $6500,  for  2  mo.  10  da.,  at  91%  ? 

12.  What  is  the  interest   of   $70.50,  for    10    years    and    10 
months,  at  5^  per  cent.  ? 

13.  What  is  the  amount  of  $45,  for  12  years  and  27  days, 
at  Gj  per  cent.  ? 

14.  What  will   $100   amount  to  in    15  years  and  6  months, 
if  put  at  interest  at  4  per  cent.  ? 

15.  How  much  will  $475.50   gain  in   5  years  9  months  and 
24  days,  at  8  per  cent.  ? 

16.  What  will  be  the  interest  of  $4560,  for  14  months  and 
19  days,  at  7  per  cent.? 

17.  What  will  $128,371   amount    to   in    10   months   and  27 
days,  at  6  per  cent.? 

18.  What  is  the  interest  of  $264.52,  for   2  years  8  months 
and  14  days,  at  6  per  cent.  ? 

19.  What  is  the   amount   of  $76.50,  for   1   year   9  months 
ind  12  days,  at  6  per  cent.  ? 

20.  What  will  be  the  interest  for  3  years  3  months  and  15 
.lays,  of  $241.60,  at  7  per  cent.? 

21.  What  is    the   interest  of  $5600,  for   30    days,  at  7  pe 
ut.? 

278.  What  is  the  rule  for  the  second  method? 


250  PERCENTAGE. 

22.  What  will  $8450  amount  to  in  60  days,  at  10  per  cent.? 

23.  What  is  the  interest  of  $4000,  for  1  month  and  6  days, 
at  9  per  cent.  ? 

24.  What  will  be  the  amount  of  88t.60,  from  September 
9th,  1852,  to  October  10th,  1853,  at  6^  per  cent.? 

25.  What  will  be  due  on  a  note  of  $126.t5,  given  July  8th 
1854,  and  payable  April  25th,  1858,  at  1  per  cent.? 

26.  What  is  the  interest  of  $350,  from  January  1st,  1856 
to  15th  of  September  next  following,  at  5 J  per  cent.? 

27.  Gave  a  note  of  $560.40,  March  14th,  1855,  on  interest, 
after  90  days  :  what  interest  was  due  December  1st,  1856,  at 
10  per  cent.  ? 

28.  What  is  the  interest  of  $1256,  for  11  months  and  9 
days,  at  6  per  cent.  ? 

29.  What  is  the  amount  of  $745.40,  at  5  per  cent,  interest 
being  reckoned  from  the  5th  day  of  the  10th  month  of  1850, 
to  the  10th  day  of  the  5th  month  of  1854  ? 

30.  September  10th,  1852,  James  Trusty  borrowed  of  Peter 
Credit  $250,  and  March  4th,  1853,  $500  more,  agreeing  to 
pay  7  per  cent,  interest  on  the  whole  :  what  was  the  amount 
of  his  indebtedness  January  1st,  1854? 

31.  Ordered  drygoods  of  A.  T.  Stewart  &  Co.,  at  different 
times,  to  the  following  amounts :  viz.,  Jan.  1st,  1854,  $254  ; 
March  15th,  1854,  $154.60  ;  April  20th,  1854,  $424.25  ;  and 
June  3d,  1854,  $75.50.  I  bought  on  time,  at  6  per  cent,  in- 
terest :  what  was  the  whole  amount  of  my  indebtedness  on 
the  first  day  of  September  following  ? 

32.  If  I  borrow  $475.75  of  a  friend  at  7  per  cent.,  wha*- 
will  I  owe  him  at  the  end  of  8  months  and  a  half? 

33.  In  settling  with  a  merchant,  I  gave  my  note  for  $127.28, 
due  in  1  year  9  months,  at  6  per  cent.  :  what  must  be  paid 
when  the  note  falls   due  ? 

34.  A  person  buying  a  piece  of  property  for  $4500,  agreed 


INTEREST.  251 

to  pay  for  it  iu  three  equal  annual  instalments,  with  interest 
at  GJ  per  cent.  :  what  was  the  entire  amount  of  money  to 
be  paid  ? 

35.  A  mechanic  hired  a  journeyman  for  9  months,  at  $40 
a  mouth,  to  be  paid  monthly  ;  at  the  end  of  the  time  he 
had  paid  nothing  ;  he  then  settled,  allowed  mterest  at  7 
l)cr  cent.,  and  gave  his  note,  on  interest,  due  in  1  year  4 
montlis  and  15  days  :  what  will  he  pay  when  his  note  falls 
due  ? 

36.  A  person  owning  a  part  of  a  woolen  factory,  sold  his 
share  for  89000.  The  terms  were,  one-third  cash,  on  delivery 
of  the  property,  one-half  of  the  remainder  in  6  months,  and 
the  rest  in  12  months,  with  7 J  per  cent,  interest  :  what 
was   the   whole   amount  paid  ? 

INTEREST    ON    NOTES. 


^^'^^•^^  Chicago,  January  Ist,  18G4. 

1.  For  value  received,  I  promise  to  pay  on  the  10th  day 
of  June  next,  to  C.  Hanford  or  order,  the  sum  of  three 
hundred  and  eighty-two  dollars  and  fifty  cents,  with  interest 
from   date,  at   t   per  cent.? 


^^^^  Baltimorb,  January  1st,  1862. 

2.  For  value  received,  I  promise  to  pay  on  the  4th  of  July, 
1864,  to  Wm.  Johnson  or  order,  six  hundred  and  twelve  dol- 
lars, with  intei'est  at  6  per  cent,  from  the  1st  of  March,  1862. 

John  Liberal. 

^^^^Q  Charleston,  July  3d,  1860. 

3.  Six  months  after  date,  I  promise  to  pay  to  C.  Jones 
or  order,  three  thousand  one  hundred  and  twenty  dollars,  with 
interest  from  the   1st  of  January  last,  at  1  per  cent. 

Joseph  Springs. 


252  PERCENTAGE. 


^'^^^•^Q  New  York  July  7tli,  1861. 

4.  Twelve  months  after  date,  I  promise  to  pay  to  Smith  & 
Baker  or  order,  seven  hundred  and  eighty-six  -f^Q  dollars,  for 
value  received,  with  interest  from  December  3d,  1861,  at  8 
per  cent.  Silas  Day. 


^^^^^'"^^  Cincinnati,  March  10th,  18G3. 

5.  Nine  months  after  date,  for  value  received,  I  promise 
to  pay  to  Redfield,  Wright  &  Co.  or  order,  four  thousand 
five  hundred  and  sixty  -^^q  dollars,  with  interest,  after  6 
months,  at  *l  per  cent.  Frederick  Stillman. 


^^^^^•^^  Boston,  July  17, 1863. 

6.  Eleven  months  after  date,  for  value  received,  we  promise 
to  pay  to  the  order  of  Fondy,  Burnap  &  Co.  one  thousand 
eight  hundred  and  fifty-four  -f^  dollars,  with  interest  from 
May  13th,   1864,  at  6  per  cent.  Palmer  &  Blake. 


279.    To  find  the  interest  of  pounds,  shillings,  and  pence. 

Rule. — I.  Reduce  the  shillings  and  pence  to  the  decimal 
of  a  pound  (Art.  214) : 

II.  Then  find  the  interest  a's  though  the  sum  were  dollars 
and  cents  ;  after  which,  reduce  the  decimal  part  of  the  answer 
to  shillings  and  pence  (Art.   215). 

Examples. 

1.  What  is  the  interest,  at  6  per  cent.,  of  £2*1  15s.  9d. 
for  2  years? 

£21  15s.  9d.  =  £21.1S16, 
j  £2tt8T5  X  .06  X  2  =  ^3.3345  interest. 

^63.3345  =  £S  6s.  S^d.  Ans. 

2.  What  is  the  interest  on  $203  18s.  6d.,  at  6  per  cent., 
for  3  years  8  months  1 6  days  ? 


INTEREST.  253 

3.  What  is  the  interest  of  ^£215  13s.  8d.,  at  6  per  cent., 
for  3  years  6  months  and  9  days? 

4.  What  is  the  interest  of  £1543  10s.  6d.,  for  2  years  and- 
a  half,  at  4  per  cent.  ? 

5.  What  is  the  amount  of  £1041  3s.,  for  1  yr.  4  mo.  15  da^ 
at  6  per  cent.  ? 

6.  What  is  the  interest  on  £511  Is.  4d.,  at  6  per  cent 
per  annum,  for  6  yr,   6  mo.  ? 

7.  What  is  the  interest  on  £161  15s.  3d.,  at  6  per  cent., 
for  8 mo.  13  da.? 

PARTIAL    PAYMENTS. 

280.  A  Partial  Payment  is  the  payment  of  a  part  of  the 
amount  due  on  a  note  or  bond.  When  partial  payments  are 
made,  they  are  indorsed  on  the  note  or  bond. 

281.  We  shall  now  give  the  rule  established  in  New  York 
(see  Johnson's  Chancery  Reports,  vol.  i.,  page  H,)  for  com- 
puting the  interest  on  a  bond  or  note,  when  partial  payments 
have  been  made.  The  same  rule  is  also  adopted  in  Massachu- 
setts, and  in  most  of  the  other  States. 

Rule. 

I.  Comjmte  the  interest  on  the  principal  to  the  time  of  thif 
first  payment ;  if  the  payment  equals  or  exceeds  this  interest^ 
add  the  interest  to  the  principal,  and  from  the  sum  subtract 
the  j^ayi^'i'ent ;  the  remainder  forms  a  new  principal : 

]  [.  But  if  the  payment  is  less  than  the  interest,  take  no 
notice  of  it,  except  to  indorse  it  on  the  note  or  bond,  until 
itther  jyayments  shall  have  been  made,  xohich  in  all,  shall 
xceed  the  interest  computed  to  the  time  of  the  last  payment ; 
(hen  add  the  interest,  so  computed,  to  the  principal,  and  from 
the  sum  subtract  the  sum  of  the  payments;  the  remainder 
will  form  a  new  principal^  on  which  interest  is  to  be  com- 
puled  as  Infore. 


254:  PEKCENTAGE. 

Examples. 
.f>349.998  Richmond,  Va.,  May  1st,  1846 

1.  For  value  received,  I  promise  to  pay  James  Wilson,  or 
order,  three  hundred  and  forty-nine  dollars  ninety-nine  cents 
mtil  eight  mills,  with  interest,  at  6  per  cent. 

James  Paywei.l. 
On  tliis  note  were  indorsed  the  following  payments  : 

Dec.  25th,  I84C,  received $49,998 

July  10th,  184T,      "  4.998 

Sept.  1st,     1848,       "  15.008 

June  14th,  1849,       "  99.999 

What  was  due  April  15th,  1850? 
Principal  on  interest  from  May  1st,  1846    -    -    -    $349,998 
Interest    to   Dec.    25th,    1846,  time    of  first  pay- 
ment, t  months  24  days 13.649 -f 

Amount $363,647  + 

Payment  Dec.  25th,  exceeding  interest  then  due  -         49.998 

Remainder  for  a  new  principal 8313.649 

Interest  of  $313,649  from  Dec.  25th,  1846,  to  June 

14th,  1849,  2  years  5  months,  19  days    -     -         46.472-1- 
Amount $360,121 


Payment,  July  10th,  1847,  less  than  in-  ,    ^ ,  ^ 
•^  '    $4,998 


terest  then  due ) 


Payment,  Sept.  1st,  1848 15.008 

Their  sum,  less  than  interest  then  due    -    $20,006 

Payment,  June  14th,  1849 99.999 

Their  sum  exceeds  the  interest  then  due  -  -  -  -  $120,005 
Remainder  for  a  new  principal,  June  14th,  1849  -  240.116 
Interest   of  $240,116    from  June   14th,  1849,    to 

April  15th,  1850,  10  months  1  day     -    -    -         12.045 

Total  due,  April  15th,  1850-    -    -    -     $252.1614- 

279.  How  do  you  find  the  interest  when  the  principal  \b  in  poiuids, 
Bhillings,  and  pence? 


INTEREST.  256 


16478.84  New  Haven,  Feb.  6th,  1850. 

2.  For  value  received,  I  promise  to  pay  William  Jenks,  or 
order,  six  thousand  four  hundred  and  seventy-eight  dollars  and 
«ighty-four  cents,  with  interest  from  date,  at  6  per  cent. 

John  Stewart. 

On  this  note  wore  indorsed  the  following  payments  : 

May  16th,  1853,  received $  545.76 

May  16th,  1855,        "         1276 

Feb.  1st,     1856,        "         2074.72. 

What  remained  due,  August  11th,  1857? 

3.  A\s  note  of  $7851.04  was  dated  Sept.  5th,  1851,  on 
Trhich  were  indorsed  the  following  payments  :  viz.,  Nov.  13th, 
1853,  $416.98  ;  May  10th,  1854,  $152  :  what  was  due  March 
Ist,  1855,  the  interest  being  6  per  cent.  ? 


$8974.56  Nj,^  York,  Jan.  3d,  1854. 

4.  For  value  received,  I  promise  to  pay  to  James  Knowles, 
or  order,  eight  thousand  nine  hundred  and  seventy-four  dollars 
and  fifty-six  cents,  with  interest  from  date  at  the  rate  of  7  per 
cent.  Stephen  Jones. 

On  this  note  arc  indoi*sed  the  following  payments  : 

Feb.  16th,  1855,  received $1875.40 

Sept.  15tli,  1856,        "         3841.26 

Nov.  11th,  1857,        "         1809.10 

June    9th,  1858,        "         2421.04. 

What  will  be  due,  July  1st,  1858? 

^345.50  BuFFAiA  Nov.  Ist,  1852. 

5.  For  value  received,  I  promise  to  pay  C.  B.  Morse,  or 
order,  three  hundred  and  forty-five  dollars  and  fifty  cents,  with 
interest  from  date,  at  7  per  cent.  John  Dob. 

i 


250  PERCENTAGE. 

On  this  note  are  the  following  indorsements  : 

June     20th,  1853,  received $75 

Jan.      12th,  1854,        "         10 

March    3d,    1855,       "        15.50 

Dec.     13th,  1856,       "         52.75 

Oct.     14th,  1857,       "        ....    -  106.75 

What  will  there  be  due,  Feb.  4th,  1858? 

^^^Q  Mobile,  Oct.  19th,  1850. 

6.  For  value  received,  we  jointly  and  severally  promise  to 
pay  Jones,  Mead  &  Co.,  or  order,  four  hundred  and  fifty  dol- 
lars on  demand,  with  interest,  at  8  per  cent. 

Mannii^g  &  Bros. 

The  following  indorsements  were  made  on  this  note  : 
Sept.    25,    1851,   received   $85.60  ;   July  10,    1852,    received 
$20  ;  June  6,  1853,  received  $150.45;  Dec.  28,  1854,  received 
$25.12J ;   May  5,    1855,   received  $169  :   what  was   due,   Oct. 
18,  1857? 

PROBLEMS    IN    SIMPLE    INTEREST. 

282.    In  every  question  of  Interest,  there   are   four  parts  : 

1st,  Principal ;    2d,  Rate  ;    3d,  Time  ;    and  4th,  Interest. 

If  any  three  of  these  parts  are  known,  the  fourth  can  be 
found.  The  interest  is  found  by  multiplying  the  principal  by 
the  rate  and  time  in  years  (Art.  276)  ;  therefore,  the  interest 
is  the  product  of  the  three  factors,  principal,  rate,  and  time. 
Any  on©  of  these  factors  is  found  by  dividing  then*  product 
by  the  other  two  :    Hence,  we   have  the  following  principles  : 

1st,  The  interest  is  equal  to  the  product  of  the  principalf 
ate,  and  time;  2d,  The  principal  is  equal  to  the  interest 
divided  by  the  product  of  rate  and  time;  3d,  The  rate  is 
equal  to  the  interest  divided  by  the  product  of  the  principal 
and  time;  4th,  The  time  is  equal  to  the  interest  divided 
hy  the  product  of  the  principal  and  rate. 


PROBLEMS  IN  INTEREST.  267 

283.    Formulaa. 
Interest  =  I  =  PxRxT. 

'  3.     11  =  JU:         4.     T=       ' 


R  X  T  '  P  X  T  '  P  X  R 

Examples. 

1.  At  what  rate  per  cent,  must  1325  be  put  at  interest 
for  1  year  and  6  months,  to   produce  an  interest   of  $34,125  ? 

Analysis. — The  product  of  principal  hj  the  time  is  325  x  1| 
=  487^.  By  principle  3d,  the  rate  equals  $34.125 -r  487.5  =  .07, 
or  7  per  cent. 

2.  What  principal,  at  6  per  cent.,  will  in  9  months  give  an 
interest  of  8178.9552? 

3.  The  interest  for  2  years  and  6  months,  at  7  per  cent., 
is   $76,965:    what   is   the  principal? 

4.  What  sum  must  be  invested,  at  6  per  cent.,  for  10 
mouths  and  15  days,  to  produce  an  interest  of  $327.3249  ? 

5.  If  my  salary  is  $1500  a  year,  what  sum  invested  at  5 
per  cent,  will  pay  it  ? 

6.  What  sum  put  at  interest  for  4  years  and  3  months, 
at  7  per  cent.,  will  gain  $283.3914? 

7.  The  interest  of  $2100  for  3  years  1  month  and  18 
days  is  $460.60:    what  is  the  rate  per  cent.? 

8.  A  person  owning  property  valued  at  $2470.80,  rents  it 
for  1  year  and  10  months  for  $452.98  :  what  per  cent,  does 
it  pay? 

9.  At  what  rate  per  cent,  must  $3456  be  loaned  for  2 
years  7  months  and  24  days,  to  gain  $503,712? 

280.  What  is  a  partial  payment  ?— 281.  What  is  the  rule  for  partial 
payments? — 283.  IIow  many  parts  arc  there  in  a  problem  of  simple 
interest?  What  are  thoy?— 283.  Write  on  the  blackboard  the  for 
mulaa  for  the  problems  of  simple  intorc>st. 


258  PERCENTAGE. 

10.  If  I  build  a  hotel  at  a  cost  of  $56000,  and  rent  it 
for  $tOOO  a  year,  what  per  cent,  do  I  receive  for  the  invest- 
nent? 

11.  The  interest  on  $1119.48,  at  1  per  cent.,  is  $195,909  : 
vhat  is  the  time? 

12  How  long  will  it  take  $500  to  double  itself,  at  6  pe. 
cut.,  simple  interest  ?  ? 

13.  Wishing  to  commence  business,  a  friend  loaned  me  $3120, 
at  6^  per  cent.,  which  I  kept  until  it  amounted  to  $5009.60  : 
Uow  long  did  I  retain  it? 

U.  I  borrowed  $700  of  my  neighbor,  for  1  year  and  8 
months,  at  6  per  cent.  ;  at  the  end  of  the  time  be  borrowed 
of  me  $750  :  how  long  must  he  keep  it  to  cancel  the  amount 
of  interest  I  owed  him  ? 

15.  What  amount  of  money  must  I  invest  at  6'/o,  that  I 
may  receive  annually  an  income  of  $450  ? 


COMPOUND    INTEREST. 

284,  Compound  Interest  is  interest  computed  on  the  amount^ 
which  is  the  sum  of  interest  and  principal  (Art.  272).  It  may 
be  computed  annually,  semi-annually,  quarterly,  monthly,  weekly, 
or  daily.  In  savings  banks,  the  interest  is  generally  computed 
semi-annually. 

Rule. —  Compute  the  interest  for  one  year,  unless  some 
other  time  is  named ;  then  add  it  to  the  principal,  and  com- 
•oute  the  interest  on  the  amount  as  on  a  new  principal ;  add 
the  interest  again  to  the  principal,  and  compute  the  interest 
as  before  ;  do  the  same  for  all  the  times  at  which  payments 
of  interest  become  due;  from  the  last  result  subtract  the 
first  principal,  and  the  remainder  ivill  be  the  compound  in- 
terest. 


284.  Wliat  is  compound  interest? 


COMPOUND    INTEREST.  259 

Examples. 

1.  What  will  be   the  compound  interest   of    $3250    for  4 
jcars,  at   7   per  cent.  ? 

OPERATION. 

$3150.000       principal  for  1st  year. 
$3750  X  .07  =       262  500       interest  for  1st  year. 
4012.500        principal  for  2(1     " 
$4012.50  X  .07  =      280.875       interest  for  2(]     " 

4293.375       principal  for  3(1     "    ^ 
$4293.375  x  .07  =      300.536  +  interest  for  3d     " 
4593.911  +  principal  for  4th    " 
$4593.911  X  .07  =      321.573  +  interest  for  4th   " 
4915.484  4-  amount  at  4  years. 
1st  principal      3750.000 
Interest  $1165.484   f 

2.  What  will   be   the    compound    interest   of    $175    for   2 
years,  at  7  per  cent.? 

3.  What  will  be  the  amount  of  $240  at  compound  interest, 
for  4  years,  at  5  per  cent.? 

4.  What    will   be   the   compound   interest   of    $300,    for    3 
years,  at  6  per  cent.  ? 

5.  What  will  be  the   compound    interest   of  $590.74,    at   6 
ptT  cent.,   for  2  years  ? 

6.  What  will  be  the  compound  interest  of  $500,  for  2  years, 
at  8  per  cent.  ? 

7.  What  will  be  the  compound   interest  of  $3758.56,  for  3 
years,  at  7  per  cent.? 

8.  What  will  be  the  compound  interest  of  $95637.50,  for  7 
years,  at  6  per  cent.  ? 

9.  What  will  be  the  compound  interest  of  $75439.75,  for  4 
years,  at  4 J  per  cent.? 


260  PERCENTAGE. 


DISCOUNT. 

285.  Discount  is  an  allowance  made  for  the  payment  oi 
in  one  J  before  it  is  due. 

The  Face  of  a  note  is  the  amount  named  in  the  note. 

The  present  value  of  a  note,  is  such  a  sum  as,  being  put 
at  interest  until  the  note  becomes  due,  would  increase  to  an 
amount  equal  to  its  face. 

The  discount,  on  a  note,  is  the  difference  between  the  face 
of  tlm  note  and  its  present  value. 

286.  Kno-wing  the  face  of  a  note,  due  at  a  future  time,  and  the 
rate  of  interest,  to  find  its  present  value. 

1.  I  give  Mr.  Wilson  my  note  for  $106,  payable  in  1  year : 
what  is  the  present  value  of  the  note,  if  the  interest  is  6  per 
cent.?     What  is  the  discount? 

Analysis. — The  present  value  operation. 

is  the  base,  the  rate  ^  is  6  per  106 

cent.,  and  the  face  of  the  note  ^^^^'  "^^^^^  "^  14.  .06  ~ 
is  the  amount  (Art.  2G7). 

Rule. — Divide  the  face  of  the  note  hy  1  dollar  plus  the 
interest  of  1  dollar  for  the  given  time. 

Note. — When  payments  are  to  be  made  at  different  times,  find  tJie 
'present  value  of  the  sums  separately,  and  their  sum  will  be  the  present 
value  of  the  note. 

Examples. 

1.  What  is  the  present  value  of  a  note  of  $615,  due  1  year 
4  months  hence,  at  t  per  cent.  ? 

2.  What  is  the  present  value  of  1202.58,  due  in  1  year  t 
months  and  18  days,  at  6  per  cent.? 

285.  What  is  discount?  What  is  the  face  of  a  note?  Wliat  is 
present  value  ?  What  is  the  discount  on  a  note  ? — 283.  Knowing  the 
face  of  a  note  and  rate,  how  do  you  find  the  present  value? 


DISCOUNT.  261 

3.  How  much  should  I  deduct  for  the  present  payment  of 
a  note  of  $721,  due  in  7  months  and  6  days,  at  5   per  cent.? 

4.  If  a  note  for  $5100  is  payable  Feb.  4th,  18G4,  what  is 
its  value  Sept.  lOtb,  1803,  interest  being  reckoned  at  8  per 
cent? 

5.  What  sum  of  money  will  amount  to  $2500,  in  2  years 
7  months  and  12  days,  at  12  per  cent.? 

G.  What  is  the  present  value  and  discount  of  $3000,  pay- 
a])le  in  1  year  2  months  and  20  days,  at  7  per  cent.? 

7.  A  held  a  note  of  $1400  against  B,  payable  Aug.  1st, 
1856  ;  B  paid  it  May  15th,  1856  :  what  sum  did  he  pay,  the 
interest  being  7  per  cent.? 

8.  A  flour  merchant  bought  for  cash  300  barrels  of  flour, 
for  $10.50  per  barrel ;  he  sold  it  the  same  day  for  $12  a 
barrel,  and  took  a  note  at  3  months  :  what  was  the  cash  value 
of  the  sale,  and  what  his  gain,  if  the  interest  is  reckoned  at 
7  per  cent.? 

9.  A  man  purchased  a  house  and  lot  for  $10000,  on 
the  following  terms  :  5000  in  cash,  2500  in  3  months,  and  the 
balance  in  six  months  :  what  was  the  cash  value  of  the  prop- 
erty, interest  being  reckoned  at  6  per  cent.? 

10.  Which  is  the  more  advantageous,  to  buy  sugar  at  7  J 
cents  a  pound,  on  4  months,  or  at  8  cents  a  pound  on  6 
months,  at  6  per  cent,  interest  ? 

11.  Bought  land  at  $10  an  acre  :  what  must  I  ask  per 
acre  if  I  abate  10  per  cent.,  and  still  make  20  per  cent,  on 
the  purchase  money? 

12.  A  merchant  owed  three  notes,  viz.,  $1000,  payable  Aug. 
1st,  1855  ;  $500,  payable  Oct.  10th,  1855,  and  $900,  payable 
Nov.  1st,  1855  :  what  was  the  cash  value  of  the  three  notes, 
"^aly  1st,  1855,  reckoning  interest  at  6  per  cent.;  and  what  was 
the  difference  between  that  value  and  their  amounts  at  the 
times  when  they  fell  due,  if  interest  were  reckoned  from  July 
Ist. 


262  PERCENTAGB. 


BANKING. 

287.  A  Corporation  is  a  collection  of  persons  authorized  by 
law  to  do  business  together.  The  instrument  which  defines 
tlieir  rights  and  powers  is  called  a  Charter, 

288.  Banks  are  Corporations  for  the  purpose  of  receiving  de^ 
posits,  loaning  money,  and  furnishing  a  paper  circulation  repre- 
sented by  specie. 

Bank  Notes  are  the  notes  made  by  a  bank  to  circulate  as 
money,  and  should  be  payable  in  specie,  on  presentation  at  the 
bank. 

A  Promissory  Note  is  the  note  of  an  individual,  and  is  a 
positive  engagement,  in  writing,  to  pay  a  given  sum,  either  on 
demand  or  at  a  specified  time. 

FORMS     «F    NOTES, 

No-  ^'  Negotiable  Note. 

^^^•^^  Providence,  May  1,  1856. 

For  value  received,  I  promise  to  pay  on  demand,  to  Abel 
Bond,  or  order,  twenty-five  dollars  and  fifty  cents. 

Reuben  Holmes. 
No-  2-  Note  Payable  to  Bearer. 

^•^^^•^^  St.  Louis,  May  1,  1855. 

For  value  received,  I  promise  to  pay,  six  months  after  date, 
to  John  Johns,  or  bearer,  eight  hundred  and  seventy-five  dol- 
lars and  thirty-nine  cents.  Pierce  Penny. 

No-  3-  2^ate  by  two  Persons. 

?^55:?1  Buffalo,  June  2,  185G. 

For  value  received,  we  jointly  and  severally  promise  to  pa;y 
to  Richard  Ricks,  or  order,  on  demand,  six  hundred  and  fifty 
nine  dollars  and  twenty-seven  cents.  Enos  Allan. 

John  Allan. 


BANKING.  263 

No.  4,  Note  Payable  at  a  Bank. 


^'20.2b  Chicago,  May  7,  1856. 

Sixty  days  after  date,  I  promise  to  pay  John  Anderson,  or 
order,  at  the   Bank  of  Commerce,  in   the   city  of  New  York, 
wcnty  dollars  and  twenty-five  cents,  for  value  received. 

Jesse   Stokes. 

Remarks  Relating  to  Notes. 

1.  The  person  wlio  signs  a  note  is  called  the  drawer  or  maker  ot 
the  note ;   thus,  Reuben  Holmes  is  the  drawer  of  note  No.  1. 

2.  The  person  who  has  the  rightful  possession  of  a  note  is  called 
the  holder  of  the  note. 

3.  A  note  is  said  to  be  negotiable  when  it  is  made  payable  to  A.  B., 
or  order,  who  is  called  the  payee  (see  No.  1).  Now,  if  Abel  Bond,  to 
whom  this  note  is  made  payable,  writes  his  name  on  the  back  of  it, 
he  is  said  to  indoi'se  the  note,  and  he  is  called  the  indorser;  and 
when  the  note  becomes  due,  the  holder  must  first  demand  payment 
of  the  maker,  Reuben  Holmes ;  and  if  he  declines  paying  it,  the  holder 
may  then  require  payment  of  Abel  Bond,  the  indorser. 

4.  When  a  note  is  not  paid  at  the  time  it  becomes  due,  the  in- 
dorser must  be  notified  of  the  fact,  and  of  the  time  it  was  due.  This 
notice  is  generally  given  by  an  officer  called  a  notary  public,  and  is 
called  a  lirotest. 

5.  If  the  note  is  made  payable  to  A.  B.,  or  bearer,  then  the  drawer 
alone  is  responsible,  and  he  must  pay  to  any  person  who  holds  the 
note. 

6.  The  time  at  which  a  note  is  to  be  paid  should  always  be  named ; 
but  if  no  time  is  specified,  the  drawer  must  pay  when  required  to  do 
so,  and  the  note  will  draw  interest  after  the  payment  is  demanded. 

7.  When  a  note,  payable  at  a  future  day,  becomes  due,  it  will  draw 
ntercst,  though  no  mention  is  made  of  interest. 

8.  In  each  of  the  States  there  is  a  rate  of  interest  established  by 
aw,  which  is  called  the  legal  interest;  and  when  no  rate  is  specified, 
he  note  will  always  draw  legal  interest.     If  a  rate  Idgher  than  legal 

.ntercst  is  named  in  the  note,  or  agreed  upon,  the  drawer,  in  most  ol 
the  States,  is  not  bound  to  pay  the  note. 

287.  What  are  corporations?    What  is  a  charter ?— 288.   What  ar© 
biuiks?    What  uro  buuk-uotos?    Wliat  is  a  promissory  note? 


261  PERCENTAGE. 

9.    If  two  i^ersons  jointly  and  severally  give  tlicir  note  (see  No.  3), 
it  may  be  collected  of  either  of  them. 

10.  The  words,  "  For  value  received,"  should  be  expressed  in  every 
note. 

11.  When  a  note  is  given,  payable  on  a  fixed  day,  and  in  a  specific 
article,  as  in  wheat  or  rye,  payment  must  be  oflfered  at  the  specified 
time ;  and  if  it  is  noc,  the  holder  can  demand  the  value  in  money. 

12.  Days  of  grace  are  days  allowed  for  the  payment  of  a  not 
after  the  expiration  of  the  time  named  on  its  face.  By  mercantile 
usage,  a  note  does  not  legally  fall  due  vmtil  3  days  after  the  expira- 
tion of  the  time  named  on  its  face,  unless  the  note  specifies  "without 
grace.'"  For  example,  No.  2  w^ould  be  due  on  the  4th  of  November, 
and  the  three  additional  days  are  called  days  of  grace. 

When  the  last  day  of  grace  happens  to  be  a  Sunday,  or  a  holiday, 
such  as  New  Year's  day,  or  the  4th  of  July,  the  note  must  be  paid 
the  day  before ;   that  is,  on  the  second  day  of  grace. 

13.  There  are  two  kinds  of  notes  discounted  at  banks :  1st.  Notes 
given  by  one  individual  to  another  for  property  actually  sold ;  these 
are  called  business  notes,  or  business  paper.  2d.  Notes  made  for  the 
purpose  of  borrowing  money,  which  are  called  accommodation  notes, 
or  accommodation  paper.  The  first  class  of  paper  is  much  preferred 
by  the  banks,  as  more  likely  to  be  paid  when  it  falls  due,  or,  in  mer- 
cantile phrase,  "when  it  comes  to  maturity." 


BANK     DISCOUNT. 

289.  Bank  Discount  is  the  deduction  made  by  a  bank  from 
the  face  of  a  note  due  at  a  future  time. 

Bunk  discount,  by  custom,  is  the  interest  of  the  face  of  the 
note,  calculated  from  the  time  when  it  is  discounted  to  the 
time  when  it  falls  due  ;  in  which  time  three  days  of  grace  are 
always  included  (see  remark  12).  The  interest  on  notes  dis 
counted  at  bank  is  always  paid  in  advance. 

The  proceeds  of  a  note  is  the  difference  between  its  face 
and  the  discount. 

289.  What  is  bank  discount?  How  is  interest  calculated?  When 
is  it  paid  ?     What  are  the  proceeds  of  a  note  ? 


BANK  DISCOUNT.  265 

290.    To  find  the  bank  discount. 

Rule. — Add  3  days  to  the  time  which  the  note  has  to  rwi, 
and  then  calculate  the  interest  for  thai  time  at  the  given  rate. 

Examples. 

1.  What  is  the  bank  discount  on  a  note  of  $300,  for  4 
Qonths,  at  6  per  cent,  per  annum. 

2.  What  is  the  bank  discount  on  a  note  of  $200,  payable 
in  five  months,  at  9  per  cent.? 

3.  What  is  the  bank  discount  and  proceeds  of  a  note  of 
$500,  at  6J  per  cent.,  payable  in  8^  months  ? 

4.  What  is  the  cash  value  of  a  note,  payable  at  bank,  of 
$1255.38,  and  due  in  4  months,  at  7  per  cent.? 

5.  What  was  the  bank  discount  on  a  note  of  $500,  due 
August  13th,  1855,  and  discounted  July  1st,  1855,  reckoning 
interest  at  7  per  cent.? 

6.  I  bought  4368  bushels  of  wheat,  at  $1.25  a  bushel,  and 
sold  it  the  same  day  for  $1.30  a  bushel  on  a  note  of  4  months. 
If  I  get  this  note  discounted,  at  bank,  at  t  per  cent.,  what  do 
I  gain  or  lose  ? 

t.  What  is  the  difference  between  the  true  and  bank  dis- 
count, of   $1000,   payable  in  7  months,  at  6  per  cent.? 

8.  What  is  the  difference  between  the  true  and  bank  dis- 
20unt,  of  $10000,  payable  in  4 J  months,  at  8  per  cent.? 

9.  January  1st,  1855,  a  note  was  given  for  $1000,  at  5 J 
per  cent.,  to  be  paid  May  1st,  next  following :  what  was  its 
cash  value  at  bank  ? 

10.  A  holds  a  note  against  B  for  $1500,  to  run  6  months 
from  Aug.  1st,  without  interest.  Oct.  1st,  he  wishes  to  pay  a 
debt  at  the  bank  of  $1000,  and  turns  in  the  note  at  a  dis- 
count of  5  per  cent,  in  payment  :  how  much  should  he  receive 
bnck  from  the  bank? 

290.  How  do  you  find  the  bank  discount? 
13 


2G6  PERCENTAGE. 

291.  To  draw  a  note  due  at  a  future  time,  whose  proceeds 
shall  be  a  given  amount. 

1.  For  what  sum  must  a  note  be  drawn  at  4  months  and 
12  days,  so  that,  when  discounted  at  bank,  at  6  per  cent.,  the 
proceeds  shall  be  $400. 

Analysis. — The  face  of  the  note  must  be  such,  that  the  interest 
for  the  given  time,  subtracted  from  the  face,  shall  leave  the  re- 
quired proceeds.  Hence,  the  proceeds  correspond  to  the  d/ifference; 
the  rate  of  interest  of  $1  for  the  given  time,  to  the  rate ;  and  the 
face  of  the  note  to  the  base  (Art.  267). 

Rule. — Divide  the  given  proceeds  hij  1  minus  the  rate  of 
%\  for  the  given  time,  and  the  quotient  will  he  the  face  of  the 
note. 

OrERATION. 

Proceeds  =  $400. 
K  %\  for  4  mo.  Zda.  =  0.0225 


_,  Proceeds 


Facc=  ^  =  $409.20 1  + 


1  -  .0225  =  0.97t5 


2.  For  what  sum  must  a  note  be  drawn  at  T  per  cent.,  pay- 
able in  6  months,  so  that  when  discounted  at  a  bank  it  shall 
produce  $285.95. 

3.  How  large  a  note  must  I  make  at  a  bank,  at  6  per 
cent.,  payable  in  6  months  and  9  days,  to  produce  $674.89? 

4.  For  what  sum  must  a  note  be  drawn,  at  5  per  cent., 
payable  in  9  months  and  15  days  after  date,  so  that  when 
discounted  at  bank,  it  shall  produce  $1000. 

5.  Marsh,  Dean  &  Co.  purchase  of  John  Jones  380  ban'cls 
of  flour,  at  $9.12|-  a  barrel,  for  which  they  give  him  a  note 
at  90  days,  for  such  sum,  that  if  discounted  at  6  per  cent.,  he 
shall  receive  the  above  price  for  his  flour  :  what  was  the  face 
of  the  note? 

291.  How  do  you  draw  a  note  whose  proceeds  shall  be  a  given 
amorint  ? 


STOCKS.  267 

STOCKS. 

292.  Capital  or  Stock  is  the  amount  of  money  paid  in  to 
carry  on  the  business  of  a  corporation. 

Stockholders  are  the  owners  of  the  stock. 
Certificates  are  the  written  evidences  of  ownership. 

293.  United  States  or  State  Stocks  are  the  bonds  of  the 
United  States,  or  of  a  State,  bearmg  a  fixed  interest. 

A  Coupon  is  a  due-bill  for  interest,  attached  to  bonds  or 
certificates  of  stock,  and  payable  at  specified  times. 

294.  Par  Value  of  stock  is  the  number  of  dollars  named  in 
each  share,  generally  100;  sometimes  50,  and  sometimes  25. 

Market  Value  of  a  stock,  is  what  it  brings  per  share,  when 
sold  for  cash. 

295.  Premium  is  the  rate  per  cent,  which  a  stock  sells  for 
above  its  par  value. 

Discount  is  the  rate  per  cent,  which  a  stock  sells  for  below 
its  par  value. 

296.  Dividend  is  a  profit  divided  among  the  stockholders, 
and  is  generally  estimated  at  a  certain  rate  per  cent,  on  the 
par  value  of  the  stock. 

297.  Brokerage  is  a  commission  made  to  an  agent  for 
buying  and  selling  stock,  uncurrent  money,  or  bills  of  exchange. 

Notes. — 1.  The  brokerage  in  the  city  of  New  York  is  generally 
ooo-fourth  of  one  per  cent,  on  the  par  value. 

2.  In  questions  of  stocks,  the  par  taZue  is  always  the  hase. 

3.  In  the  examples,  the  shares  are  |100  each,  unless  another  amount 
is  named. 

298.  To  find  the  dividend  on  a  given  amount  of  stock. 

1.  What  is  the  percentage  on  25  shares,  $100  each,  of  Kings 
County  Insurance  Company,  the  dividend  being  25  per  cent.? 

Analysis. — Here,  the  base  and  rate  oxq  given  to  find  the  per- 
centage (Art.  259). 


268  PERCENTAGE. 

2.  The  Atlantic  Fire  Insurance  Co.  declares  a  semi-annual 
dividend  of  4-|%  on  the  capital  stock:  what  is  the  annual 
dividend  of  43  shares  at  that  rate? 

3.  The  Atlantic  Bank  of  Brooklyn  has  declared  a  serai^ 
annual  dividend  of  6%:   what  is  the  dividend  on  18  shares? 

4.  A  bankrupt  is  indebted  to  A,  $5416,  and  to  B,  |6t95 
what  does  each  receive,  when  the  dividend  to  the  creditors  i 
47 J  per  cent.? 

5.  A  mining  company,  shares  $25  each,  declared  a  dividend 
of  n"/o:    what  was  the  dividend  on  36  shares? 

299.  To  find  the  value  of  stock  which  is  above  or  below 
par. 

1.  What  is  the  value  of  $5600  of  stock,  reckoned  at  par, 
when  the  stock  is  at  a  premium  of  9  per  cent.  ? 

Analysis. — In  this  class  of  examples,  the  base  and  rate  are 
given.  When  the  stock  is  above  par,  the  amount  is  required  (Art. 
265) ;   when  it  is  below  par,  the  difference  (Art.  2C6). 

2.  What  is  the  cost  of  56  shares  of  !N'ew  York  Central 
Railroad  stock,  at  5  J  per  cent,  below  par,  and  the  brokerage 
J  per  cent.  ? 

3.  I  bought  36  shares  in  the  Pennsylvania  Coal  Company, 
at  a  discount  of  12 J  per  cent.,  and  sold  them  at  a  premium 
of  7  per  cent.,  paying  j-  per  cent,  brokerage  in  each  case  :  how 
much  did  I  make  by  the  operation? 

4.  What  is  the  market  value  of  216  shares  of  bank-stock, 
each  share  $15,  and  the  premium  t|"/o  ? 

292.  Wliat  is  capital  or  stock  ?  Who  are  stockholders  ?  What  are 
certificates?  —  293.  Wliat  are  United  States  stocks?  What  is  a 
coupon? — 294.  What  is  the  par  value  of  a  stock?  What  is  market 
value?— 295.  What  is  premium?  Wliat  is  discount ?— 296.  What  is 
dividend  ? — 297.  What  is  brokerage  ?  W^liat  is  the  general  rate  in  the 
city  of  New  York?  What  is  the  base,  in  stocks? — 298.  How  do  you 
find  the  dividend  on  stock  ? — 299.  How  do  you  find  the  value  of  stock 
when  it  is  above  or  below  par? 


I 


STOCKS.  2C9 

5.  The  par  value  of  257  sliares  of  bank-stock  is  $200  a 
l.arc  :  what  is  tlie  present  value  of  ail  tlic  shares,  the  stock 
)  irii^  at  a  premium  of  15  per  cent.? 

0.  What  is  the  value  of  120  shares  of  Exchange  Bank 
!ock,  it  being  at  a  premium  of  18|  per  cent.,  and  the  par 
;i]ue  being  $150  a  share? 

7.  What  will  be  the  cost  of  69  sliares  of  Panama  Railroail 
lock,    at   a   discount   of   8"'o,  the   par  value   being  |125,   and 

in'okeragc  }  per  cent.? 

8.  Gilbert  &  Co.  buy  for  Mr.  A,  200  shares  of  United 
States  stock,  at  a  premium  of  GJ  per  cent.,  and  charge  -J-  per 
cent,  brokerage  :  if  the  sliares  are  $1000  each,  how  much 
money  does  A  pay  for  the  stock? 

9.  Mr.  B.  bought  125  shares  of  stock  in  the  American 
Guano  Company,  at  par,  the  shares  being  $20  each.  At  the 
end  of  4  months,  he  received  a  dividend  of  5  per  cent.,  and  at 
the  end  of  10  months,  a  second  dividend  of  4  per  cent.  At 
the  end  of  the  year,  he  sold  his  stock  at  a  premium  of  10  per 
cent.  :  how  much  did  he  make  by  the  operation,  reckomng  the 
interest  of  money  at  7  per  cent.? 

300.  To  find  how  much  stock,  at  par  value,  a  given  sum  of 
money  will  purchase,  when  the  stock  is  at  a  premium  or  discount. 

i  .  1.  What  value  of  stock,  at  par,  can  be  purchased  by 
t$3045.38,  if  the  stock  is  at  a  premium  of  10  per  cent.,  and 
}  per  cent,  is  charged  for  brokerage  ? 

Analysis. — When  the  stock  is  above  par,  the  amount  and  the 
rate  are  given  to  find  the  base  (Art.  207);  when  below  par,  the 
difference  and  rate  are  given  to  find  the  base  (Art.  267). 

2.  A  person  wishes  to  invest  $3000  in  bank-stock,  which  is 
at  a  discount  of  15  per  cent.  :  what  amount  at  par  can  he 
purchase  ? 

300.  Hovr  do  you  find  how  much  stock,  at  par,  a  given  sum  of  money 
will  buy  when  the  stock  is  at  a  preini um  ?    How  when  it  is  at  a  discount  ? 


270  PERCENTAGE. 

3.  How  many  shares  of  Galena  and  Chicago  Ra,iIroad  stock 
can  be  bought  for  $6384,  at  14%  premium? 

4.  When  bank-stock  sells  at  a  discount  of  t^%,  what 
amount  of  stock,  at  par  value,  will  $3700  buy? 

5.  A  person  has  $7000,  which  he  wishes  to  invest  ;  what 
will  it  purchase  in  5  per  cent,  stocks,  at  a  discount  of  3J  per 
cent.,  if  he  pays  \  per  cent,  brokerage  ? 

6  How  much  6  per  cent,  stock,  at  par,  can  be  purchased 
for  $8700,  at  SJ  per  cent,  premium,  i  per  cent  being  paid  for 
brokerage  ? 

7.  A  person  owning  $12000  in  government  funds,  desires  to 
purcliase  stock  in  the  American  Exchange  Bank.  The  funds 
are  at  a  discount  of  3^  per  cent.,  while  the  bank-stock  is  at  a 
premium  of  10^  per  cent. :  what  amount  of  stock,  at  par  value, 
can  he  purchase,  allowing  the  broker's  charges  for  the  purchase 
to  be  f  per  cent.? 

301.  To  find  the  rate  of  interest  on  an  investment  in  stock, 
when  the  stock  is   above  or  below  par. 

1.  What  is  the  rate  of  interest  on  an  investment  in  6  per 
cent,  stocks,  when  they  are  at  a  discount  of  25  per  cent.? 

Analysis. — The  interest    on    the  opkeation. 

stock  is  computed  on  its  par  value;  .75  )  _  <^i    ^    Ofi  —  ^  Ofi 

the  interest  on  the    investment    is  ic  J  ~              .       —     . 
computed  on  the  market  value,  and 

the  percentage  in  each  case  is  the  ^  — -  _2   :::::   Qg 

same.     Hence,  1  dollar  of  the  stock  ''^ 
multiplied  by  its    rate  of   interest, 

will  be  equal  to  the  market  value  ^'*^-  ^  ^^  ^^^^• 
©f  $1   of  the  stock  multiplied  by  its  rate  of  interest. 

Rule. — 3IuUiply  $1  o/*  the  stock  by  its  rate  of  interest,  and 
divide  the  product  by  the  mai'ket  value  of  %\  of  the  stock: 
the  quotient  will  be  the  rate  of  interest  on  the  investment. 

2.  If  I  buy  7  per  cent,  stock  at  12J  per  cent  discount, 
B»^at  is  the  rate  per  cent,  on  the  investment? 


s'i'ooKs.  271 

3.  If  the  stock  of  the  Erie  Raih-oad  sells  at  62J  per  cent., 
nud  pays  semi-annual  dividends  of  2^  per  cent.,  what  would  be 
the  rate  of  interest  on  an  investment  ? 

4.  The  bonds  of  the  Illinois  Central  Railroad  Company, 
which  bear  interest  of  1  per  cent.,  are  worth  87  per  cent.,  and 
the  charge  for  brokerage  is  J  per  cent.  :  what  would  be  the 
interest  on  an  investment  in  these  funds  ? 

5.  The  stock  of  the  Hartford  and  New  Haven  Railroad  is 
at  a  premium  of  20  per  cent. :  reckoning  the  interest  on  money 
at  6  per  cent.,  what  will  be  the  interest  on  an  investment? 

302.  To  find  how  much  a  stock  must  bo  above  or  below 
par,  to  produce  a  given  rate  of  interest. 

I.  At  what  rate  must  a  6  per  cent,  stock  be  bought,  so 
that  the  investment  shall  yield  9  %  interest  ? 

Analysis.  —  Since    the    per-  operatiox. 

centage   is   the    same    in  both  ^   I  =  Si  X    06  =  $  06 

cases,  $1  of  the  stock  multiplied  .09  f 
by  its  rate  of  interest,  is  equal 

to  the   market   value  of  $1  of  x  =  - —  =  .66J 

the    investment    multiplied    by  * 

^  ^^^-  1  -  Mi  =  .331  dis. 

Rule. — I.  Multiply  %1  of  the  stock  by  its  rate  of  interest, 
and  divide  the  product  by  the  rate  of  interest  on  the  invest- 
ment: the  quotient  ivill  be  the  per  cent,  of  the  market  value 
of  ^l  of  the  stock: 

II.  If  the  market  value  is  greater  than  1,  subtract  1  from 
it,  and  the  remainder  will  be  the  per  cent,  of  premium ;  if 
less  than  1,  subtract  it  from  1,  and  the  remainder  will  be  the 
per  cent,  of  discount, 

301.  How  do  you  find  the  rate  of  interest  on  an  investment  v/licn  tlio 
Block  is  above  or  below  par  ? — 302.  How  do  you  find  how  miicli  a  stock 
must  be  above  or  below  par,  to  produce  a  given  rate  of  interest  ? 


272  PERCENTAGE. 

2.  At  what  rate  of  discount  must  I  invest  in  8  per  cent, 
stock,  in  order  to  yield  me  10  per  cent.? 

3.  If  the  par  value  of  a  stock  is  $100,  and  the  interest  t 
per  cent.,  what  is  the  discount  when  an  investment  yields  13 
per  cent.? 

4.  At  what  rate  must  I  invest  in  a  9%  stock,  that  I  maj 
receive  8  per  cent,  on  my  investment? 

303.  Which  is  the  best  investment? 

1.  I  invest  $1250  in  State  stocks  bearing  aj  interest  of  6 

per  cent.,  and  a  premium  of  15  per  cent.     I  invest  an  equal 

amount  in  State  fives   at   12  per  cent,  discount.    Which  will 
yield  the  larger  interest  ? 

Analysis. — Find  the  rate  of  interest  of  each  investment,  and 
then  compare  the  two  rates.  That  investment  which  produces  the 
greater  rate  is  the  more  advantageous. 

operation. 
1st.  2d. 


^^=.0521==5,V.V. 


.88 


05 

_  =  .0568  =  5/o%°/<' 


The   second   investment  is   the   more   advantageous. 

2.  Which  is  the  better  investment,  to  buy  sixes  at  par,  or 
sevens  at  107  ? 

3.  Which  will  yield  the  larger  profit,  8  per  cent,  stock  at 
a  premium  of  20  per  cent.,  or  5  per  cent,  stock  at  80  per 
cent.  ? 

4.  If  I  invest  $2000  in  State  stocks  at  5  per  cent.,  at 
par,  and  an  equal  amount  at  6  per  cent.,  at  90,  what  will 
be  the  difference  of  the  proceeds  of  the  investments  at  the 
tmd  of  5  years? 

303.  How  do  you  determine  which  is  the  best  investment  ? 


INSURANCE.  273 


INSURANCE. 

304.  An  Insurance  Company  is  a  company  chartered  to 
iiLsure  against  risks. 

Insurance  is  an  indemnity  for  loss  or  injury.     It  is  made  by 
iiipauies  or  individuals,  in  consideration  of  a  certain  sum  paid. 

Underwriters  or  Insurers  are  the  companies  or  persons  who 
insure. 

305.  Insurance  is  now  limited,  chiefly,  to  three  classes  of 
i.-asi'S  : 

1.  Fire  insurance,  or  insurance  against  loss  by  fire. 

2.  Marine  insurance,  or  insm-ance  against  loss  by  water. 

0.  Life  insurance,  or  insurance  against  loss  by  death. 

306.  A  Mutual  insurance  is  one  in  which  the  insured  share 
in  tlie  profits. 

307.  A  policy  is  the  mutual  agreement  of  the  parties. 

308.  Premium  is  the  percentage  paid  by  him  who  owns  the 
[)roperty  to    him  who   insures   it,  as   a   compensation  for  risk. 

309.  All  the  cases  of  insurance  are  simple  applications  of 
tlie  principles  of  percentage.     There  are  four  : 

1.  To  find  the  premium,  when  the  base  and  rate  are  known 
(Art.  259). 

•2.  To  find  the  rate,  when  the  base  and  premium  arc  known 
(Art.  201). 

3.  To  find  the  base,  when  the  rate  and  premium  are  known 
(Art.   2C2). 

4.  To  find  the  percentage,  when  the  premium  is  insured  as 
well  as  the  base.  The  base  insured  is  then  the  premium  plus 
the  first  base. 


304.  Wliat  is  an   insurimce   company?     What  is  insurance?    Who 
are  luidtnvr iters? 


274  PERCENTAGE. 

Examples. 

1.  What  would  be  the  premium  for  insuring  a  ship  and 
cargo,  valued  at  $147614,  at  3^  per  cent.? 

2.  What  would  be  the  insurance  on  a  ship,  valued  at 
$47520,  at  J  of  1  per  cent.     At  ^  of  1  per  cent.? 

3.  What  would  be  the  insurance  on  a  house,  valued  at 
116800,  at   IJ  per  cent.?     At  f  of  1  per  cent.? 

4.  A  merchant  owns  |  of  f  of  a  ship,  valued  at  $24000, 
and  insures  his  interest  at  2  J  per  cent. :  what  does  he  pay  for 
his  policy? 

5.  What  will  it  cost  to  insure  a  store,  worth  $5640,  at  | 
per  cent.,  and  the  stock,  worth  $7560,  at  f  per  cent.? 

6.  A  carriage-maker  shipped  15  carriages,  worth  $425  each: 
what  must  he  pay  to  obtain  an  insurance  upon  them  at  75 
cents  on  a  hundred  dollars  ? 

7.  A  merchant  imported  150hhd.  of  molasses,  at  35  cents 
a  gallon;  he  gets  it  insured  for  3i  per  cent,  on  the  selUng 
price  of  50  cents  a  gallon  :  if  the  whole  should  be  destroyed, 
and  he  get  the  amount  of  insurance,  how  much  would  he  gain  ? 

8.  If  I  get  my  house  and  furniture,  valued  at  $3640,  insured 
at  4J  per  cent.,  what  would  be  my  actual  loss  if  they  were 
destroyed  ? 

9.  The  ship  Astoria  was  valued  at  $20450,  and  her  cargo 
at  $25600,  and  was  bound  on  a  voyage  from  New  York  to 
Canton.  The  vessel  was  insured  at  the  St.  Nicholas  office  for 
$12000,  at  2f  per  cent.,  and  the  cargo  for  $18500,  at  3}  per 
cent.  The  vessel  foundered  at  sea  :  what  was  the  entire  loss 
of  the  owner? 

10.  Shipped  from  New  York  to  the  Crimea,  5000  barrels  of 


305.  To  how  many  classes  is  insurance  limited?  What  are  they? 
—306.  What  is  mutual  insurance ?— 307.  What  is  tlie  policy?— 308. 
What  is  premium ?— 309.  How  many  cashes  are  there  of  insurance? 
What  are  they? 


'our,  worth  $10.50  a  barrel.     The  premuim  paid  was  $2887.50  : 
vhat  was  the  rate  per  cent,  of  the  hisurance? 

11.  Paid  $120  for  insurance  on  my  dwelling,  valued  at 
17500  :  what  was  the  rate  per  cent.? 

12.  A  merchant  imported  225  pieces  of  broadcloth,  each 
)iece  containing  40  yards,  at  $3.50  a  yard:  he  paid  $1323  for 
nsurance  :  what  was  the  rate  per  cent.? 

13.  A  merchant  paid  $1320  insurance  on  his  vessel  and 
cargo,  which  was  5^-  per  cent,  on  the  amount  insured  :  how 
much  did  he  insure? 

14.  A  man  pays  $51  a  year  for  insurance  on  his  storehouse, 
at  IJ  per  cent.,  and  $126.45  on  the  contents,  at  2J  per  cent.  : 
what  amount  of  property  does  he  get  insured  ? 

15.  A  person  shipped  15  pianos,  valued  at  $275  each.  He 
insures  them  at  3  per  cent.,  and  also  insures  the  premium  at 
the  same  rate  :  what  insurance  must  he  pay  ? 

IG.  A  store  and  its  contents  are  valued  at  $16750.  The 
owner  insures  them  at  IJ  per  cent.,  and  then  insures  the 
premium  at  the  same  rate  :  what  insurance  must  he  pay  ? 


LIFE    INSURANCE. 

310.  Life  Insurance  is  an  agreement  to  pay,  in  consideration 
of  a  premium,  a  specified  amount  to  parties  named  in  the 
agreement,  in  case  of  the  death  of  the  party  insured. 

311.  To  enable  the  company  to  fix  their  premiums  at  such 
rates  as  shall  be  both  fair  to  the  insured  and  safe  to  the 
association,  they  must  know  the  average  duration  of  life  from 
any  given  time  to  its  probable  close.  This  average  is  called 
the  "Expectation    of   Life,"   and   is    determined    by   collecting 

310.  "What  is  life  insurance? — 311.  What  is  necessary  to  enable  a 
company  to  fix  their  preniiuma?    How  is  the  expectation  dt'terniinedlf 

^''  '    '     '     :  '      '       '      ,    •■      .  xpccUUiim  ol    lilcV 


276  PERCENTAGE. 

from  many  sources  the  most  authentic  information  in  regard  to 
the  average  duration  of  life  from  any  period  named. 

If  we  take  100  infants,  some  will  die  in  infancy,  some  in 
childhood,  and  some  in  old  age.  It  has  been  found,  from  care- 
ful observation,  that  if  the  sum  of  their  ages,  after  the  last 
shall  have  died,  be  divided  by  100,  the  quotient  will  be  38.'I2 
very  nearly  :  hence  38.t2  is  said  to  be  the  "  Expectation  of 
Life"  at  infancy. 

The  Carlisle  Tables,  which  are  used  in  this  country  and  Eng- 
land, show  the  "Expectation  of  Life"  from  1  to  100  years 
At  10  years  old  it  is  found  to  be  48.82  ;  at  20,  41.46  ;  at 
30,  it  is  34.34  ;  at  40,  2t.61  ;  at  50,  it  is  21.11  ;  at  60,  14 
years  ;  at  tO,  9.19  ;  at  80,  5.51  ;  at  90,  3.28  ;  and  at  100, 
it  is  2.28  years. 

If  we  wish  the  expectation  of  life,  between  the  periods 
named  in  the  table,  we  can  readily  find  it  by  the  rules  of  pro- 
portion. Thus,  if  we  wished  the  expectation  of  life  at  16  years, 
we  should  observe  that,  at  10  years,  it  has  been  found  to  be 
48.82  ;  at  20  years,  it  has  been  found  to  be  41.46  ;  hence,  for 
10  years  it  varies   48.82  —  41.46  =  7.36  years  : 

Then,         10     :     6     :  :     t.36     :     4.416; 

which   number  being   subtracted  from   48.82,  leaves  44.40,  the 
expectation  of  life  at  16  years  of  age. 

312.  From  the  above  facts,  and  the  value  of  money  (which 
is  shown  by  the  rate*  of  interest),  a  company  can  calculate  with 
great  exactness  the  amount  which  they  should  receive  annually, 
for  an  insurance  on  a  life  for  any  number  of  years,  or  during 
its  entire  continuance. 

Among  the  principal  life  insurance  companies  in  the  United 
States,  are  the  New  York  Life  Insurance  and  Trust  Company 
the  Girard  Life  Insurance,  Annuity  and  Trust  Company  Oi 
Philadelphia,  and  the  Massachusetts  Hospital  Life  Insurance 
and  Trust  Company  of  Boston.  The  rates  of  insurance,  in 
tliose  companies,  differ  but  little. 


INSURANCE  277 

313.  All  companies  have  published  tables  which  show  the 
quarterly,  semi-annual,  and  annual  premiums  that  must  be  paid 
on  each  $100  or  $1000  insured. 

Note. — Experience  has  demonstrated  that  the  risks  are  about 
equal  on  all  ages  between  14  and  25  years.  Persons  under  the  ago 
of  25  years  are  charged  for  wJiole  life  policies,  the  rate  at  that 
ago;  though  dividends  are  based  on  the  true  age.  An  extra  charge, 
on  the  above  rates,  of  one-half  per  cent,  on  the  amount  insured,  is 
made  for  insuring  the  lives  of  women  under  the  age  of  48  years. 

Examples. 

1.  A  person,  20  years  of  age,  finds  that  the  premium,  per 
annum,  is  $1.36  on  8100  :  what  must  he  pay  to  insure  his  life 
for  1  year  for  $8950  ? 

2.  A  man,  aged  40  years,  wishes  to  insure  his  life  for  5 
years,  and  finds  that  the  annual  rate  is  $1.86  for  $100  :  how 
much  premium  must  he  pay  per  annum  on  $12500? 

3.  A  person,  38  years  of  age,  obtains  an  insurance  on  his 
life  for  5  years,  at  the  rate  of  $1.75  per  annum  on  $100  : 
ho\f  much  is  the  annual  premium  on  $15000  ? 

4.  A  person  going  to  Europe,  expecting  to  retura  in  2 
year»,  effects  an  insurance  on  his  life  at  J  of  J  per  cent, 
premium  on  $100  ;  he  insures  for  $5000  :  what  is  the  annual 
premium  ? 

5.  What  will  be  the  annual  premium  for  insuring  a  person's 
life,  who  is  60  years  of  age,  for  $2000,  at  the  rate  of  $4.91 
on  $100? 

6.  A  person,  at  the  age  of  50  years,  obtained  an  insurance 
at  4J  per  cent,  per  annum  on  each  $100 ;  he  insured  for 
^1500,  and  died   at  the  age  of  70.     How  much  more  was  the 

nsurance  than  the  payments,  without  reckoning  interest  ? 

7.  A  gentleman,  47  years  of  age,  going  to  China  as  ambas- 
sador, obtains  an  insurance  on  his  life  for  $10000,  by  paying  a 
premium  of  $2,71  per  annum  on  every  $100,  and  dies  at  tlie 
liiidiile  of  tlic  third  year :  reckoning  simple  interest  on  liLs 
payments  vit  7  per  cent.,  what  is  gained  by  the  insurance  ? 


27S 


PERCENTAGE. 


ENDOWMENTS. 

314.  An  Endowment  is  a  certain  sum  to  be  paid  at  the 
expiration  of  a  given  time,  in  case  the  person,  on  whose  life  it 
is  taken  shall  live  till  the  expiration  of  the  time  named. 

Tlie  following   table   shows   the  value  of  an  endowment  pnr- 

liiised    for    $100,    at  the    several    periods    mentioned    in    the, 

column    of   ages,    the    endowment    to    be    paid   if   the    person 

attains  the  age  of  21  years.     The  table  is  calculated  under  the 

hypothesis  that  money  is  worth  6  per  cent,  interest. 

TABLE     OF     ENDOWMENTS, 

Showing  the  sum  to  be  paid  at  21  years,  if  alive. 


Age. 

Birth $376.84 

3  months 344.28 

6        "      331.40 

9        "      ....  318.90 

1  year 30G.58 

2  "    271.03 

3  "    243. G9 

4  "    225.42 


Age. 

5  years. . 

..$210.53 

6      "     .. 

..     198.83 

7     "     .. 

..     188.83 

8     "     .. 

..     179.97 

9      "     .. 

..     171.91 

10     "     . . 

..     164.46 

11      "     . . 

..     157.43 

12     "     . . 

..     150.64 

Age. 

13  years $144.12 

86 
.83 

97 
.31 


137. 
131 
125. 
120. 
114. 
109. 
104. 


This  table  shows  that  if  $100  be  paid  at  the  birth  of  a 
child,  he  will  be  entitled  to  receive  $376.84,  if  he  hves  to 
attain  the  age  of  21  years.  If  $100  be  paid  when  he  is  ten 
years  old,  he  will  be  entitled  to  receive  $164.46,  if  he  lives  to 
attain  the  age  of  21  years.  And  similarly  for  other  ages.  We 
can  easily  find  by  proportion, 

1st.  How  much  must  be  paid,  at  any  age  under  21,  to  pur- 
chase a  given  endowment  at  21  ;    and, 

2d.  What  endowment  a  sum  paid  at  any  age  under  21,  will 
purchase. 

Examples. 

1.  Wliat  endowment,  at  21,  can  be  purchased  for  $250,  paid 
at  the  age  of  10  years? 

2.  What  endowment,  at  21,  can  be  purchased  for  $360,  paid 
at  the  age  of  5  ye:irs  ? 


ANNUITIES. 


279 


3.  If  my  child  is  T  years  old,  and  I  purchase  an  endowineni 
for  $650,  wiiat  will  he  receive  if  he  attains  the  age  of  21 
years  't 


ANNUITIES. 

315.  An  Annuity  is  a  fixed  sum  of  money  to  be  paid  at 
regular  periods,  generally,  yearly,  either  for  a  limited  tune,  or 
forever,  in   consideration  of  a  given  sum  paid  in  hand. 

The  Present  Value  of  an  annuity  is  that  sum  which,  being 
put  at  compound  interest,  would  produce  the  sums  necessary  to 
pay  the  annuity. 

Tiie  purchaser  of  an  annuity  should  pay  more  than  the  com- 
pound interest ;  for  the  seller  cannot  afford  to  take  the  money 
of  the  purchaser,  invest  it,  reinvest  the  interest,  and  pay  over 
the  entire  proceeds. 

Knowing  the  rate  of  interest  on  money,  and  the  present 
value  of  an  annuity,  a  close  estimate  may  be  made  of  the  price 
it  ought  to  sell  for. 

Table, 

Skoiciiiff  the  pitESKNT  VALUK  OF   AN   ANNUITY  OF  $1,   /roiH  1   to  30  ijcars,    at 
dijferent  rates  of  interest . 


Tears. 

5  percent 

6  per  cent. 

Years. 

5  per  cent 

6  per  cent 

1 

0.052.*]81 

0.943390 

16 

10.837770 

10.105895 

2 

1.8r)JJ410 

1.833393 

17 

11.274060 

10.477260 

3 

2.72:3248 

2.073012 

18 

11.689587 

10.827603 

4 

3.r)45i)50 

3.405106 

19 

12.085321 

11.158116 

5 

4.329477 

4.212304 

20 

12.462216 

11.469921 

6 

5.075692 

4.917324 

21 

12.821153 

11.764077 

7 

5.78G373 

5.582381 

22 

13.163003 

12.041582 

8 

0.403213 

0.209794 

23 

13.488574 

12.303379 

9 

7.107822 

6.801092 

24 

13.798642 

12.550358 

10 

7.721735 

7.360087 

1  25 

14.093945 

12.783356 

11 

8.300414 

7.886875 

I  26 

14.375185 

13.003166 

12 

8.803252 

8.388844 

27 

14.043034 

13.210534 

13 

9.393573 

8.852083 

28 

14.898127 

13.400164 

U 

9.808(541 

9.294984 

29 

15.141074 

13.590721 

15 

10  379058 

9.712249 

30. 

15.372451 

13.764831 

280  PERCENTAGE. 

To  find  the  present  value  of  an  annuity  for  any  rate,  and 
for  any  time,  we  simply  multiply  the  present  value  of  an  an- 
nuity of  $1  for  the  same  rate  and  time,  by  the  annuity,  and 
the  jDroduct  will  be  its  present  value. 

Thus,  the  present  value  of  an  annuity  of  $600  for  8  years, 
t  6  per  cent.,  is 

$6,209^94  X  600  -  $3725.8'764  ;  that  is, 

pres.  val.  of  $1  x  annuity  =  pres.  val. ;  hence, 

.,  pres.  val.  ^.       _ 

annuity  =  — - — - — -^-j-  ;  therefore, 
^        pres.  val.  of  $1 '  ' 

316.  To  find  what  sum  will  produce  a  certain  annuity  at  a 
given  rate   and  for   a   given  time. 

Rule. — Maltiphj  the  present  value  of  an  annuity  of  $1, 
at  the  given  rate  and  for  the  given  time,  by  the  given  annu- 
ity;   the  product  will  he  that  sum. 

317.  To  find  what  annuity  a  given  sum  will  produce  at  a 
given  rate,   and  for   a  given   time. 

R/Ule. — Divide  the  given  sum,  or  present  valuer  by  the  pres- 
ent value  of  $1,  for  the  given  rate  and  time,  and  the  quotient 

will  be  the  annuity. 

Examples. 

1.  What  is  the  present  value  of  an  annuity  of  $550,  at  5 
per  cent.,  for  21  years? 

2.  What  would  be  the  value  of  an  annuity  that  should  yield 
eight  hundred  and  thirty-five  dollars  a  year  for  sixteen  years, 
the  interest  being  compound,  and  at  the  rate  of  5  per  cent, 
per  annum  ? 

3.  What   is    the    present   value   of  an  annuity   of  $1500  a 
year,  for  30  years,  the   compound  interest  being  reckoned  at 
per  cent.  ? 

314.  What  is  an  endowment  ?  Wliat  does  the  table  of  endo^^^nent8 
show  ?  What  may  be  foui^d  from  the  table  ? — 315,  What  is  an  annu- 
ity? What  is  tlie  present  vWue  of  an  annuity? — 316.  How  do  you 
find   the  present  value  of  an   annuity  *or  a  given  rate   and    time? 


ASSESSING  TAXES.  281 

4.  What  annuity,  for  twenty-four  years,  could  be  purchased 
for  the  sura  of  twenty-seven  thousand  five  hundred  and  sixty 
dollars,  the  compound  interest  being  reckoned  at  6  per  cent.  ? 

5.  Mr.  Jones  having  a  small  fortune  of  $25000,  and  calcu- 
lating that  he  would  live  about  20  years,  purchased  an  annuity 
ai  C  per  cent.,  with  an  agreement  that  he  would  pay  $20  a 
)  car  to  an  invalid  sister :  what  was  his  annual  income  from 
the  investment  after  making  that  payment? 


ASSESSING    TAXES. 

318.  A  Tax  is  a  certain  sum  required  to  be  paid  by  the  in- 
habitants of  a  town,  county,  or  State,  for  the  support  of  gov- 
ernment. It  is  generally  collected  from  each  individual,  in 
proportion  to  the  amount  of  his  property. 

In  some  States,  however,  every  white  male  citizen,  over  the 
age  of  twenty-one  years,  is  required  to  pay  a  certain  tax. 
This  tax  is  called  a  poll-tax  ;  and  each  person  so  taxed  is 
called  a  poll. 

319.  In  assessing  taxes,  the  first  thing  to  be  done  is  to 
make  a  complete  inventory  of  all  the  property  in  the  town,  on 
which  the  tax  is  to  be  laid.  If  there  is  a  poll-tax,  make  a 
full  list  of  the  polls,  and  multiply  the  number  by  the  tax  on 
each  poll,  and  subtract  the  product  from  the  whole  tax  to  be 
raised  by  the  town  ;  the  remainder  will  be  the  amount  to  be 
raised  on  the  property.  This  remainder  is  the  percentage  or 
tax  to  be  raised.  The  value  of  the  property  taxed  is  the  base  ; 
hence   this   remainder,    divided   by  the  value   of  the    property, 

317.  How  do  you  find  what  annuity  a  given  sum  will  produce,  at 
a  given  rate  and  for  a  given  time? — 318.  What  is  tax?  IIow  is  it 
generally  collected?  What  is  a  poll-tax? — 319.  What  is  the  first 
tiling  to  be  done  in  assessing  a  tax  ?  If  there  is  a  poll-tax,  hov/  do  you 
find  tlie  amount?  IIow,  then,  do  you  find  the  per  cent,  of  tax  to  be  levied 
on  a  dollar?    IIow  do  you  find  the  tax  to  bo  raisf^d  on  each  individual? 


2S2  rEROKNTAGE. 

gives   the  rate.     Each   man's  property,  multiplied  by  the  rate, 
gives  his  tax  or  percentage. 

Examples. 

A  certain  town  is  to  be  taxed  $4280  ;  the  property  on 
which  the  tax  is  to  be  levied  is  valued  at  $1000000.  Now 
there  are  200  polls,  each  taxed  $1.40.  The  property  of  A  is 
valued  at  $2800,  and  he  pays  4  polls. 


B's  at  $2400,  pays  4  polls, 
C's  at  $2530,  pays  2     " 
D's  at  $2250,  pays  6     " 


E's  at  $7242,  pays  4  polls, 
F's  at  $1651,  pays  6     " 
G's  at  $1600.80,  "     4     " 


What  will   be    the   tax   on   one   dollar,   and  what  will   be  A*s 
tax;  and,  also,  that  of  each  on  the  list? 

First,         $1.40  X  200  =    $280,  amount  of  poll-tax. 

$4280  —  $280  =  $4 000,  amount  to  be  levied  on  property. 
Tlien,  $4000  -^  $1000000  =  .004  =  j^U  =  4  mills  on  $1. 

Now,  to  find  the  tax  of  each,  as  A's,  for  example : 

A's  inventory, $2800 

.004 


$11.20 

4  polls,  at  $1.40  each, 5.60 

A's  whole  tax, $16.80 

In  the  same  manner,  the  tax  of  each  person  in  the  town- 
ship may  be  found. 

Examples. 

1.  In  a  county  embracing  350  polls,  the  amount  of  property 
on  the  tax-list  is  $318200  ;  the  amount  to  be  raised  is  as  fol- 
lows :  for  State  purposes,  $1465.50  ;  for  county  purposes, 
$350.25  ;  and  for  town  purposes,  $200.25.  By  a  vote  of  the 
county,  a  tax  is  levied  on  each  poll  of  $1.50  :  how  much  per 
c^nt.  will  be  laid  upon  the  property  ? 

2.  In  a  county  embracing  a  pojxilation  of  98415  persons,  a 


ASSESSING    TAXES.  2S3 

tax  is  levied  for  town,  county,  and  State  purposes,  amounting 
to  $100400.  Of  this  sum,  a  part  is  raised  by  a  tax  of  25 
cents  on  each  poll,  and  the  remainder  by  a  tax  of  two  mills 
on  the  dollar :  what  is  the   amount  of  property  taxed  ? 

3.  In  a  county,  embracing  a  population  of  56450  persons, 
a  tax  is  levied  for  town,  county,  and  State  purposes,  amount 
ing  to  |!874Gt  ;  the  personal  and  real  estate  is  valued  a 
$4890300.  Each  poll  is  taxed  25  cents  :  what  per  cent,  is  the 
lax,  and  how  much  will  a  man's  tax  be,  who  pays  for  five 
polls,  and  whose  property  is  valued  at  $5400  ? 

What  is  B's  tax,  who  is  assessed  for  2  polls,  and  whose 
property  is  valued  at  $3760.50? 

4.»A  banking  corporation,  consisting  of  40  persons,  was 
taxed  $957.50 ;  their  property  was  valued  at  $125000,  and 
each  poll  was  assessed  50  cents  :  what  per  cent,  was  their  tax, 
and  what  was  a  man's  tax,  who  paid  for  1  poll,  and  whose 
share  was  assessed  for  $2000? 

5.  What  sum  must,  be  assessed  to  raise  a  net  amount  of 
$5674.50,  allowing  2J  per  cent,  commission  on  the  money  col- 
lected ? 

6.  Allowing  4  per  cent,  for  collection,  what  sura  must  be 
assessed  to  raise  $21346.75  net? 

7.  In  a  certain  township,  it  becomes  necessary  to  levy  a 
tax  of  $4423.2475,  to  build  a  public  hall.  The  taxable  prop- 
erty is  valued  at  $916210,  and  the  town  contains  150  polls, 
each  of  which  is  assessed  50  cents.  What  amount  of  tax  must 
be  raised  to  build  the  hall,  and  pay  5  per  cent,  for  collection, 

ud  what  Ls  the  tax  on  a  dollar? 

What  is  a  person's  tax  who  pays  for  3  polls,  and  whose 
j>ersonal  property  is  valued  at  $2100,  and  his  real  estate  at 
«3000  ? 

What  is  G's  tax,  who  is  assessed  for  1  poll,  and  $1275.50? 

What  is  n's  tax,  who  is  asscssoxl  for  1  poll,  and  $2456  ? 


284  EQUATION  OF  PAYMENTS. 

8.  The  people  of  a  school  district  wish  to  build  a  new 
school-liouse,  which  shall  cost  $2850.  The  taxable  property  of 
the  district  is  valued  at  $190000  :  what  will  be  the  tax  on  a 
dollar,  and  what  will  be  a  man's  tax.  whose  property  is  valued 
at  $7500? 

How  much  is  Mr.  Merchant's  tax,  whose  personal  and  real 
estate  are  assessed  for  $1200? 

9.  In  a  school  district,  a  school  is  supported  by  a  rate-bill. 
A  teacher  is  employed  for  6  months,  at  $60  a  month  ;  the  fuel 
and  other  contingencies  amount  to  $66.  They  drew  $41.60 
public  money,  and  the  whole  number  of  days'  attendance  was 
1688  :  what  was  D's  tax,  who  sent  148  days  ? 

What  was  F's  tax,  who  sent   184^  days? 


EQUATION  OF   PAYMENTS. 

320.  Equation  of  Payments  is  the  process  of  finding  the 
average  time  of  payment  of  several  sums  due  at  dififerent  times, 
so  that  no  interest  shall  be  gained  or  lost. 

The  average  or  equated  time,  is  the  time  that  elapses  from 
the  time  at  which  we  begin  to  reckon  interest  to  the  time  of 
payment  of  all  the  debts.  The  equated  date  is  the  date  of 
payment  of  all  the  debts. 

321.  When  the  times  of  payment  are  reckoned  from  the 
same   date. 

1.  B  owes  Mr.  Jones  $5t  :  $15  is  to  be  paid  in  6  months  ; 
$18  in  T  months  ;  and  $24  in  $10  months  :  what  is  the  average 
time  of  payment,  so  that  no  interest  shall  be  gained  or  lost  ? 

Analysis. — The  interest   of   $15   for    6  opehation. 

months,  is  the  same  as  the  interest  of  $1  $15  X     Q  =     90 

for  90  months;    the  interest  of  $18  for  7  $18  X     7  =  126 

mouths,  is  the  same  as  the  interest  of  $1  &24  x  10  =  240 

for   126   months;    and  the  interest  of  $24  7^            ^l^dT^fR 

ibr  10  months,  is  the  same  as  the  interest  456 

of  $1  for  240  months;    hence,  the  sura  of  


EQUATION   OF   PAYMENTS.  285 

these  products,  456,  is  the  number  of  months  it  would  take  $1  to 
produce  the  required  interests.  Now,  the  sum  of  the  payments,  $57, 
will  produce  the  same  interest  in  one  fifty -seventh  part  of  the  time; 
tliat  is,  in  8  months:  hence,  to  find  the  average  time  of  payment: 

Rule. — Multiply  each  j^ayment  by  the  time  before  it  becomes 
due,  and  divide  the  sum  of  the  2)roducts  by  the  sum  of  the 
vaymenfs:  the  quotient  will  be  the  average  time. 

Examples. 

1.  A  merchant  owes  $1200,  of  which  $200  is  to  be  paid  in 
4  months,  $400  in  10  months,  and  the  remainder  m  16  months : 
if  he  pays  the  whole  at  once,  at  what  time  must  he  make  the 
payment  ? 

2.  A  owes  B  $2400  ;  one-third  is  to  be  paid  in  6  months, 
one-fourth  in  8  months,  and  the  remainder  in  12  months :  what 
is  tlic  mean  tunc  of  payment? 

3.  A  merchant  has  due  him  $4500  ;  one-sixth  is  to  be  paid 
in  4  months,  one-third  in  6  months,  and  the  rest  in  12  months: 
what  is  the  equated  time  for  the  payment  of  the  whole  ? 

4.  A  owes  B  $1200,  of  which  $240  is  to  be  paid  in  three 
months,  $360  in  five  months,  and  the  remainder  in  10  months  : 
wliat  is  the  average  time  of  payment  ? 

5.  Mr.  Swain  bought  goods  to  the  amount  of  $3840,  to  be 
paid  for  as  follows,  viz.  :  one-fourth  in  cash,  one-fourth  in  6 
months,  one-fourth  in  7  months,  and  the  remainder  in  one  year: 
what  is  the  average  time  of  payment? 

6.  A  man  bought  a  farm  for  $5000,  for  which  he  agreed 
to  pay  $1000  down,  $1200  in  3  months,  $800  in  8  months, 
$1500  in  10  months,  and  the  remainder  in  one  year:  if  he  pa}s 
the  whole  at  once,  what  would  be  the  average  time  of  pay 
ment  ? 

320.  Wliat  is  equation  of  payment«?  What  is  the  average  or 
equated  time? — 321.   How  do  you  find  equated  time? 


2S6  EQUATION   OF   FAYMEXTS. 

*l.  A  person  owes  three  notes  :  the  first  is  for  $200,  pay- 
able July  1st ;  the  second  for  $150,  payable  August  1st ;  and 
the  third  for  $250,  payable  August  15th :  what  is  the  average 
time,  reckoned  from  July  1st? 

322.    When  the  times   are  reckoned  from  different  dates. 

1  E.  Bond, 

Bought  of  Trust  &  Co. 

1861.     Aug.    1,  450  yds.  muslin,  at  10  cents    -    -  -    $45  00 

Aug.  16,  800  yds.  calico,  at  12J  cents  -    -  -     100  00 

Sept.    5,  720  yds.  bombazine,  at  80  cents  -  -     576  00 

Oct.      1,  300  yds.  cloth,   at   $3.50    -    -    -  -  1050  00 

On  what  day  may  the  whole  be  supposed  to  have  been  pur- 
chased ;    or,  what  is  the  equated  date  of  purchase  ? 

Analysis. — The  owner  parted  with  his  goods,  and  therefore 
with  their  values,  at  the  dates  specified;  and  the  question  is,  to 
find  at  what  time  he  could  have  sold  the  whole  at  the  same 
advantage.  Keckoning  from  Aug.  1st,  the  earliest  date,  he  had  the 
use  of  $45,  the  amount  of  the  first  sale,  for  no  time ;  of  $100  for 
15  days,  viz.,  from  Aug.  1st  to  Aug.  IGth;  of  $576  for  35  days, 
viz.,  from  Aug.  1st  to  Sept.  5th;  of  $1050  for  61  days,  from 
Aug.  1st  to  Oct.  1st:  then,  by  the  preceding  Article,  we  have 
the  following  operation: 

OPERATION. 


45 

100 

576 

1050 

X 
X 
X 

X 

0 
15 
35 
61 

= 

000 

1500 

20160 

64050 

1771 

1771)85710(483-Wt^ 

7084 

=  equated  1 

!;ime. 

14870 
14168 

48  days 
equated 

from  Aug. 
date,  Sept. 

1st. 
18th. 

702 

1771 

322.  From  what  date  may  the  equated  time  be  reckoned  ?  What  is 
the  multiplier  of  the  date  used  as  the  point  of  departure?  What  do 
you  do  when  the  quotient  contains  a  fraction?  What  is  the  rule 
when  the  times  are  reckoned  for  different  dates? 


KQU. n'lON    OF    PAYMENTS.  287 

Instea<l   of    reckoning    from    tho    earliest  date,   we  might  have 
reckoned   backward  from   the   latest  date. 

From  Oct.  1st  to  Sept.  5th   there  are  26  days, 
to  Aug.  16th         "         46      " 
to  Auff.  1st  "         61      " 


*o* 


45  X  61  =    2745 

100  X  46  =    4600 

576  X  26  =  14976 

1050  X     0  =    0000 

1771  22321  -^  1771  =  l^^i  days. 

13  days  from  Oct.  1st  or  Sept.  18th. 

Rule. — I.  From  the  date  assumed  as  the  point  of  reckon- 
ing, find  the  intervening  days  to  each  date,  and  multiply  each 
amount  by  its  number  of  days: 

II.  Divide  the  sum  of  these  products  by  the  sum  of  the 
payments,  and  the  quotient  will  be  the  equated  time  in  days. 
This  number,  reckoned  from  the  assumed  date,  will  give  the 
equated  date. 

Note. — 1.  The  equated  time  may  be  reckoned  from  the  earliest  or 
any  previous  date,  or  from  the  latest  date  or  any  date  subsequent  to  it 

2.  The  multiplier  of  tho  date  used  as  the  point  of  reckoning  is  0, 
and  the  corresponding  product  is  nothing.  The  payment  must  be 
added,   in   finding  the  sum  of  the  payments. 

3.  When  the  fraction  in  the  quotient  is  less  than  \,  it  is  rejected 
when  greater  than  i,  1  is  added  to  the  days. 

2.  Mr.  Johnson  sold,  on  a  credit  of  8  months,  the  following 

bills  of  goods  : 

April  1st,  a  bill  of  14350, 

May  7th,   a  bill  of     3750, 

June  5th,   a  bill  of     2550 

A.t  what  time  will  the  whole  become  due  ? 

Note. — Find  the  equated  date  oi  purchase,  to  wliich  add  the  time 
of  credit;  if  the  times  of  credit  vary,  find  the  times  of  payment, 
and  then  equate. 

3,  A  purchased  of  B  the  following  bill  of  goods,  on  different 
times  of  credit : 


288  EQUATION    OF   PAYMENTS. 

May  1st,    1851,  a  bill  amounting  to  $800  on  3  months. 

June  1st,       "  ''               "  "      TOO    "   3 

"     15th     "  "                "  "      900    "   4        " 

July  25th,     '*  "               "  "    1000    "    6 

What  is  the  equated  time  for  the  payment  of  the  whole,  and 
Qu  what  day,  reckoned  from  Aug.  1st,  is  the  bill  due? 

4.  A  person  purchased  the  following  bills  of  goods,  on  dif- 
ferent times  of  credit  : 

Jan.    1st,  1855,  a  bill  amounting  to  $36t.20  on  4  months. 

"      28th,  "         "  "            "      901.80    "   3 

Feb.    24th,  "         "  "            "      826.38    "   5       " 

March  30th,  "         "  "            "      854.88    "   6 .      " 

May    1st,  "         "  "            "      396.50    "   4       " 

What  is  the    average  time  of  payment  from  the  time  the  first 
bill  falls  due  ?     On  what  day  is  the  payment  made  ? 

5.  A  flour  merchant  bought  at  one  time  150  barrels  of 
flour,  at  $8  a  barrel ;  15  days  afterward  he  bought  176  bar- 
rels, at  $8.50  a  barrel ;  25  days  after  that  he  bought  200 
barrels,  at  $9  a  barrel :  how  many  days  after  the  first  pur- 
chase would  be  the  equated  time  of  payment? 

323.  To  find  how  long  a  sum  of  money  must  be  at  interest 
to  balance  the  interest  on  a  given  sum  for  a  given  time. 

1.  If  A  lends  B  $100  for  3  months,  how  long  ought  B  to 
lend  A  $500  to  balance  the  interest  ? 

Analysis. — Since    $700,    in    3  months,  operation. 

will  produce  as  much  interest  as  $2100  in       700  X  8  =  2100  ; 

1  month,  it  will  require  as  many  months       2100  -^  500  =  4^ 

for  $500  to  produce  the  same  interest,  as  j        n  .i 

.^^  .  ..-,..         .     ^.^^  -d?i8.  41  months 

500  IS  contanied  times  m  2100.  "* 

2.  A  lends  B  his  note  for  $900,  payable  in  5  months  :  for 
what  length  of  time  should  B  lend  to  A  his  note  for  $480,  to 
balance  the  favor? 


EQUATION    OF  PAYMENTS.  28^ 

3.  C  buys  of  D  100  barrels  of  flour,  at  $tj  per  barrel, 
and  in  payment  gives  Iiis  note  for  3  months  ;  D  buys  of  C 
500  bushels  of  wheat  at  80  cents  per  bushel,  and  gives  his 
note  in  payment :  how  long  must  this  note  run,  that  each  may 
hare  an  equal  use  of  the  other's  money? 

324.  To  find  how  long  the  balance  may  be  kept,  when  pay- 
nents   are  made  before  they   are  due. 

1.  A  owes  B  $800,  payable  in  6  months  ;  at  the  expiration 
of  4  months,  he  pays  $500  :  how  long  beyond  the  6  months 
should  A  retain  the  balance,  so  that  neither  shall  make  or  lose 
interest  ? 

Analysis. — A  has    the    right    to  retain  operation. 

the  $800  for    6    months,   or  $4800  for  1  800  X  6  =  4800 

month.    He  retains  $500  for  4  months,  or  500  X  4  =  2000 

$2000  for  1  month.     Hence,  he  may   still  300                2800 

retain  $2800  for  1  month,  or  the  balance,  ^oaq  _^  onn  _  qi 

$300,  as  many  months  as  SOO  is  contained  *          ~~    ^ 

thnes   in   $2800;    or,   9}    months  from  the  date  of  the  debt;    or, 
9  J  —  6  =  3  J  months  beyond  the  time  of  six  months. 

2.  C  owes  D  $2500,  payable  in  4  months  ;  but  at  the  end 
of  3  months  pays  him  $1600:  how  long  after  the  payment  of 
$1600  should  the  remainder  be  retained  to  balance  the  account? 

3.  One  merchant  owes  another  $1600,  payable  in  6  months, 
but  at  the  end  of  3  months  pays  $400  ;  at  the  end  of  4 
months  $400,  and  at  the  end  of  $5  months  $300  :  how  long, 
from  the  last  payment,  may  the  balance  be  retained  to  square 
the  account? 

4.  Mr.  Jones  owes  his  grocer  $900,  due  9  months  from  the 
1st  day  of  January  ;  June  15,  paid  $520  :  on  what  day  was 
the  remainder  due? 

5.  A  note  for  $500,  dated  November  6th,  1856,  payable  in 
3  months,  was  given  by  E  to  F.  On  December  3d,  E  paid 
$350  :    at  what  time  was  the  balance  due  ? 

13 


290  EQUATION  OF  PAYMENTo- 

325.    To  find  the  cash  balance  of  an  account. 

An  account  is  said  to  be  balanced  when  the  sum  of  the 
items  on  the  debit  side  is  equal  to  the  sum  of  the  items  on 
the  credit  side.  When  these  two  sums  are  unequal,  such  an 
amount  is  added  to  the  less  as  will  make  the  sum  equal  to 
the  greater.  This  is  called  the  balance.  There  are  three  kinds 
of  balances  : 

1st.  The  merchandise  balance ,  in  which  interest  on  the  items 
is  not  considered  ; 

2d.  The  interest  balance,  which  adjusts  the  interest  on  the 
two  sides  of  an  account ;  and 

3d.  The  cash  balance,  which  arises  from  combining  the  mer- 
chandise balance  with  the  interest  balance. 

Accounts  are  settled  either  by  cash  or  by  note.  In  ascer- 
taining the  cash  balance  of  an  account,  interest  is  allowed  on 
all  the  items  of  both  sides  ;  the  balance  of  interest  makes  a 
new  item,  and  may  belong  to  either  side  of  the  account. 

Ascertain  the  cash  balance  of  the  following  account  on  the 
25th  of  April,  1850  : 

Dr.  S.   SxoDGRAss.  Or. 


1850.    Aprillst, To  goods,  $375.00 

"17th,   "      "         268.00 

"25th,   "      *♦         175.00 

Cash  balance,  237.93 

$1055.93 


April  7th,  By  goods,  $675.00 
"  15th,  "  "  380.00 
"      25th,  Bal.  of  Int., ^ 

$1055.93 


Analysis. — Reckoning  backwards  from  April  25th,  we  find  the 
days  for  which  we  charge  interest,  and  these  are  used  as  multipliers. 
The  interest  of  $375  for  24  days  is  the  same  as  the  interest  of  $1  for 
9000  days ;  and  so  of  the  other  items.  The  difference  of  the  sums 
of  these  products,  is  the  number  of  days  which  $1  must  be  at  interest 
to  produce  the  balance  of  interest,  and  the  balance  always  goes  with 
the  larger  sum  of  the  products. 

325.  Wliat  is  the  rule  for  finding  the  cash  balance  of  an  account  ? 


EQUATION   OF   PAYMENTS.  291 

Debtor  Items.  operatiox.  Creditor  Bems. 

375  X  24  =  9000 

268  X     8  =  2144 

175  X     0  =  0000 

11144 


675 

X 

18  = 

12150 

380 

X 

10  = 

3800 
15950 
11144 

4806 

0.93  +, 

balance  of  interest 

Then,         4806  x  ^% 

Rule. — I.  Take  the  latest  date  of  the  account,  or  any  later 
dale  at  which  the  balance  is  to  be  struck,  as  the  point  of 
reckoning,  and  find  the  days  between  this  date  and  the  date 
of  each  item  ;  and  consider  these  days  as  multipliers : 

II.  Multiply  each  item  by  its  multiplier;  then  take  the 
difference  of  the  sums  of  these  products,  and  midtiply  it 
hij  the  interest  for  one  day:  the  result  will  he  the  interest 
balance,  ichich  is  to  be  added  to  the  side  having  the  greater 
sum : 

III.  Tfien  find  the  cash  balance. 

Notes. — 1.  If  the  cash  balance  had  been  required  on  any  day  aftei 
the  2oth  of  April,  the  mode  of  proceeding  would  have  been  exactly  the 
same. 

2.  In  the  examples,  the  rate  of  interest  will  be  taken  at  7  per  cent., 
and  360  days  in  the  year. 

3.  After  the   balance  of  interest  is  found,  the  cash  balance  is  ob- 
tained by  adding  the  two  sides  of  the  accoimt,  taking  the  differenc 
of  the  sums,  and  placing  it  on  the  smaller  side  of  the  account. 

4.  If  the  cash  balance  is  settled  by  a  note,  interest  should  run  o 
he  note  from  the  date  of  the  cash  balance  to  the  time  of  payment. 

5.  Let  the  pupil  find  the  interest  and  cash  balance  in  each  of  th 
lilowing  examples: 

2.  What  is  the  balance  of  interest,  and  what  the  cash  bal- 
ance, on  the  following  account,  on  March  20  th  ? 


EQUATION   OF  PAYMENTS. 


Dr, 

S. 

Johnson. 

Cr. 

56.  Jan.  1,  To  merch., 

$500 

Jan.   5,     Bj  cash. 

$850 

"   16,    "  cash, 

450 

"  19,      "     merch., 

780 

Feb.  5,    "  merch., 

680 

"   25,      " 

250 

a    24,    "         " 

800 

Feb.  15,     "     cash, 

600 

Mar.  1,   "  cash. 

150 

Cash  balance, 

700.5' 

"  16,    "  merch., 

600 

Interest  balance. 

.58 

fe9,fiftn.'sft 

-^9>fiR0..^ft 

Note. — 1.  When  the  items  have  the  same  or  different  times  of 
credit  allowed,  find  when  the  items  are  payable,  and  then  proceed 
as  before. 

2.  If  the  cash  balance  is  required  on  a  day  previous  to  the  latest 
date  of  the  items,  find  the  cash  balance  for  this  latest  date ;  then  find 
the  present  value  for  the  given  date :    this  wiU  be  the  cash  balance. 

3.  Allowing  a  credit  of  six  months  on  each  item,  what  is 
the  interest  and  cash  balance,  Feb.  1st,  1856? 


Dr. 


R.  Sherman. 


Gr, 


1855.  July  1st,  To  merch.,  $750 
"  17th,  "  "  600 
"  25th,  "        "  800 


$2150 


Feb.  6th,  By  merch., 
Mar.  7th,    "      " 

Interest  balance. 
Cash  balance, 


900 
46.20 
403.80 
$2150.00 


4.  Allowing  a  credit  of  3  months  on  each  of  the  items  of 
the  following  account,  what  would  be  the  interest  and  cash 
balance  on  October  31st,  1856  ? 


Dr. 


E,.  Rivers. 


856.  May  1,   To  merch.,  $500 

"  20,     "         "  675 

June  6th,  To  cash,      350 

July  9th,  "  merch.,  175 

Cash  balance,        620.70 


$2320.70 


Or. 


May   6th,     By  cash, 
"    25th,      "  mer.. 

$400 
620 

June  16th,     "  cash. 

900 

July  20th,      "  mer.. 
Interest  balance, 

400 
.70 

$2320.  70 

i 


EQUATION   OF  PAYMENTS.  293 


326.  To  find  the  equated  time  of  settling  an  account  con- 
taining debtor  and  creditor  items. 

To  equate  an  account,  is  to  fix  the  time  of  payment  of  the 
ulurcliandise  bahmce  in  such  manner  that  the  interest  of  each 
side  shall  be  equal.  The  object  of  equating  accounts  is  two- 
1  ill :  1st.  To  find  for  what  time  interest  must  be  charged  ou 
file  balance  ;  2d.  To  find  the  date  of  a  note,  whose  running 
time  is  fixed,  and  which  is  given  in  payment  of  the  balance. 

Properly,  the  face  of  the  note  should  be  the  sum,  whose 
present  value  is  the  balance. 

If  the  note  is  given  without  interest,  then  its  face  is  the 
balance  ;  and  if  the  note  becomes  payable  before  the  latest 
date,  then  interest  must  be  charged  for  the  remaining  time. 
The  process  of  equating  accounts  is  similar  to  that  of  finding 
the  cash  balance  :    hence,  we  have  the  following 

Rule.  — I.    Find   the   merchandise  balance: 

II.  Find  the  number  of  days  between  the  latest  date  of 
cither  side,  and  the  date  of  each  item,  and  consider  these 
numbers  as  InultipHers : 

III.  MuUiiily  each  item  by  its  multiplier ;  then  take  the 
difference  of  the  sums  of  these  products  and  divide  it  by  the 
merchandise  balance:  the  quotient  is  the  number  of  days, 
which,  carried  backward  or  forward  from  the  latest  date^ 
will  give  the  equated  date. 

Note. — When  the  greater  sura  of  the  items  and  the  greater  sum  of  the 
products  fall  on  the  same  side  of  the  account,  the  quotient  is  to  he  car- 
ried backward  from  the  latest  date  :  and  forward,  when  these  sums  ar« 
found  on  different  sides. 

1.   Equate  the  following  account : 

Dr.  James  '. 

1801.  Jan.    16,  To  merch.,  $716.75 

"      25,   "       "  000.00 

Feb.     7,   "       "         2705.50 

Mar.  19,   "  cash  701.25 


N 

. 

Cr. 

1. 

Jan. 

19, 

By  cash. 

$500.15 

Feb. 

.1, 

"    merch. 

,    1015.25 

Mar. 

7, 

"    cash. 

1200.00 

April 

3, 

"    merch. 

,     712.00 

294 


EQUATION   OF  PAYMENTS. 


2.   What  is  the  balance  of  the  folio winff  account — when  due? 


Dr. 


Israel  Jenkins. 


!835.  May    6,  To  merch.,  $7150.00 

"      16,   "        "  475.00 

June  17,   "       "         3475.25 

"    21,  "       "         1516.50 

July    5,  "       "  279.00 


Gr. 
$2450.00 


1835.  May    9,  By  cash, 

"      21,   "      "  915.00 

June  12,   "  merch.,  4165.50 

"     19,   "  merch.,  2915.50 


3.   What    is   the   equated  date  for  the  payment   of  the  bal- 
ance of  the  following  account  ? 

Dr.  Jacob  Parton.  Cr, 


1861.  June    6,  To  merch.,  $8000.00 

"    23,  "       "  1756.50 

"     30,  "    cash,        2890.75 

July  12,  "    note,        3000.15 


1861.  June    2,  By  merch.,  $7450.75 

"     19,  "        "  2695.25 

July  10,  "        "  1865.50 

"      16,  "        "  970.00 


827.  Account  of  Sales. 

An  Account  of  Sales  is  an  account  of  the  goods  sold,  with 
their  prices,  the  charges  thereon,  and  the  net  proceeds.  Such 
an  account  a  consignee  transmits  to  the  consignor.  The  net 
proceeds  of  such  sale  is  nothing  but  the  cash  balance,  due  at 
the  equated  date.    We  will  illustrate  by  the  following  example : 


ACCOUNT   OF    SALES    OF    FLOUll    FOR    A.    MATr^E^YS,    CHICAGO. 


Date. 

Purchaser. 

Description. 

Price. 

1863. 
Nov.    5, 
Dec.     6, 
Dec.  19, 
Dec.  23, 

James  Jackson, 
Robert  Fisk, 
Francis  Sutton, 
James  Lyon, 

75  bbls.  superfine, 
89  bbls.  Excelsior, 
120  bbls.  fine, 
66  bbls.  ordinary, 

350  bbls. 

$6.90 
7.20 
6.30 
5.90 

$517.50 
640.80 
756.00 
389.40 

$2303.70 

CHARGES. 

Nov.  10th,  cash  paid  transportation, 

Nov.     6th,  insurance, 

Dec.    23d,  storage, 


Commission  on  $230'\70,  at  2^%, 
Total,    ■ 


$16 
9 

10 

57.58 
$152.58 


ALLIGATION.  295 

ALLIiS-ATION. 

328.  Alligation  is  the  process  of  mixing  substances  in  such 
a  manner  that  the  value  of  the  compound  shall  be  equal  to  the 
sum  of  the  values  of  the  several  ingredients.  It  is  divided  into 
two  parts  :     Alligation  Medial  and  Alligation  Alternate. 

ALLIGATION   MEDIAL. 

329.  Alligation  Medial  is  the  method  of  finding  the  price 
or  quality  of  a  mixture  of  several  simple  ingredients  whose 
prices  and  quantities  are  known. 

I.  A  grocer  would  mix  200  pounds  of  lump  sugar,  worth 
13  cents  a  pound,  400  pounds  of  Havana,  worth  10  cents  a 
pound,  and  600  pounds  New  Orleans,  worth  T  cents  a  pound : 
what  should  be  the  price  of  the  mixture? 

Analysis.— The  quantity,  2001b.,  opebation. 

at  13  cents  a  pound,  costs  $2G;  400  200  X  13  =  26.00 
poimds,  at  10  cents  a  pound,  costs  400  X  10  =  40.00 
$40;  and  GOO  lb.  at  7  cents  a  pound,  600  X  1  =  42.00 
costs  $42:    hence,  the  entire  mix-       1200  )  108.00(9  cts. 

ture,    consisting    of    1200  lb.,   costs 

$108.  Now,  the  price  of  the  mixture  will  be  as  many  cents  as 
1200  is  contained  times  in  10800  cents,  viz.,  9  times:  hence,  to 
find  the  price  of  the  mixture, 

Rule. — I.   Find  the  cost  of  the  mixture : 

II.  Divide  the  cost  of  the  mixture  by  the  sum  of  the 
simples,  and  the  quotient  will  be  the  price  of  the  mixture. 

Examples. 

1.  If  1  gallon  of  molasses,  at  75  cents,  and  3  gallons,  at 
60  cents,  be  mixed  with  2  gallons,  at  31  J,  what  is  the  mixture 
worth  a  gallon? 

2.  If  teas  at  3TJ,  50,  62i,  80,  and  100  cents  per  pound,  be 
mixed  together,  what  will  be  the  value  of  a  pound  of  the 
mixture  ? 


296  ALLIGATION. 

3.  If  5  gallons  of  alcohol,  worth  60  cents  a  gallon,  and  3 
gallons,  worth  96  cents  a  gallon,  be  diluted  by  4  gallons  of 
water,  what  will  be  the  price  of  one  gallon  of  the  mixture  ? 

4.  A  farmer  sold  50  bushels  of  wheat,  at  $2  a  bushel ;  60 
bushels  of  rye,  at  90  cents  ;  36  bushels  of  corn,  at  62 J  cents ; 
and  50  bushels  of  oats,  at  39  cents  a  bushel :  what  was  the 
average  price  per  bushel  of  the  whole? 

5.  During  the  seven  days  of  the  week,  the  thermometer 
stood  as  follows  :  70°,  73°,  13^°,  17°,  70°,  80^°,  and  81°  : 
what  was  the  average  temperature  for  the  week  ? 

6.  If  gold  18,  21,  17,  19,  and  20  carats  fine,  be  melted 
together,  what  will  be  the  fineness  of  the  compound? 

7.  A  grocer  bought  341b.  of  sugar  at  5  cents  a  pound, 
1021b.,  at  8  cents,  1361b.  at  10  cents  a  pound,  and  341b.  at 
12  cents  a  pound.  He  mixed  it  together,  and  sold  the  mix- 
ture so  as  to  make  50  per  cent,  on  the  cost  :  what  did  he 
sell  it  for  per  pound  ? 

8.  A  merchant  sold  81b.  of  tea,  111b.  of  coffee,  and  251b. 
of  sugar,  at  an  average  of  15  cents  a  pound.  The  tea  was 
worth  30  cents  a  pound  ;  the  coffee,  25  cents  a  pound  ;  and 
the  sugar  7  cents  a  pound  :  did  he  gain  or  lose,  and  how 
much? 

ALLIGATION    ALTERNATE. 

330.  Alligation  Alternate  is  the  method  of  finding  what 
proportion  of  several  simples,  whose  prices  or  qualities  are 
known,  must  be  taken  to  form  a  mixture  of  any  required 
price  or  quality.  It  is  the  reverse  of  Alligation  Medial,  and 
may  be  proved  by  it. 

The  process  of  Alligation  Alternate  is  founded  on  an  equality 
of  gain  and  loss.  In  selling  a  mixture  at  an  average  price, 
there  is  a  gain  on  each  simple  below  that  price,  and  a  loss  on 
each  simple  above  that  price.  The  gain  must  be  exactly  equal 
to  the  loss,  otherwise  the  7alue  of  the  compound  would  not  be 
an  average  price. 


ALTERNATE. 


297 


CASE     I. 

331.    To  find   the  proportional  parts. 

1.    A  miller  would  mix  wheat,  worth    12  shillings  a  bushel ; 

corn,  worth  8  shillings  ;    and  oats,  worth   5   shillings,  so  as  to 

nmke   a  mixture   worth   T    shillings   a  bushel :     what   are   the 

pioportional  parts  of  each  ? 


OPERATION. 


oats. 


5s.- 

7s.  -jcorn,        8s.- 
( wheat,   1 2s. 


A. 

B. 

c. 

D. 

i 

4 

5 

1 

1 

2 

i 

2 

B. 
6  or  3 
2    "   1 
2    "   1 


Analysis. — On  every  bushel  put  into  the  mixture,  whose  price 
is  leKS  than  the  mean  price,  there  will  be  a  gain;  on  every 
bushel  whose  price  is  greater  than  the  mean  price,  there  will  be 
a  loss ;  and  since  the  gains  and  losses  must  balance  each  oilier, 
we  must  connect  an  ingredient  on  which  there  is  a  gain  with 
<me  on   which   there   is   a  loss. 

A  bushel  of  oats,  when  put  into  the  mixture,  will  bring  7 
shillings,  giving  a  gain  of  2  shillings;  and  to  gain  1  shilling,  we 
must  take  lialf  as  nmch,  or  ^  a  bushel,  which  we  write  opposite  53. 
in  column  A. 

On   1    bushel   of  wheat  there  will  be  a   loss  of  5  shillings;  and 


we  write  in  column  A:  ^  and  },  are  called  proportional  numbers. 

Again,  comparing  the  oats  and  corn,  there  is  a  gain  of  2  shil- 
lings on  every  bushel  of  oats,  and  a  loss  of  1  shilling  on  every 
bushel  of  corn:  to  gain  1  shilling  on  the  oats,  and  lose  1  shil- 
ling on  the  corn,  we  must  take  i  a  bushel  of  the  oats,  and  1 
bushel  of  the  corn:  these  numbers  are  written  in  column  B. 
Two  simples,  thus  compared,  are  called  a  couplet:  in  one,  the 
price  of  1  is  less  than  the  mean  price,  and  in  the  other  it  is 
great^. 

If  every  time  we  take  ^  a  bushel  of  oats  we  take  \  of  a 
bushel  of  wheat,  the  gain  and  loss  will  balance ;  and  if  every 
time  we  take  \  a  bushel  of  oats  we  take  1  bushel  of  corn,  the 
gain    and   loss  will  balaice:    hence,   if  the   proportional    numberi 


298  ALLIGATION. 

of  a   couplet    he    multiplied    ty  any    numher,    the  gain  and  loss 
denoted  hy    the  products  will  halance. 

When  the  proportional  numbers,  in  any  column,  are  fractional 
(as  in  columns  A  and  B),  multiply  them  by  the  least  common 
multiple  of  their  denominators,  and  write  the  products  in  new 
columns  0  and  D.  Then  add  the  numbers  in  columns  0  and  D 
standing  opposite  each  simple,  and  if  their  sums  have  a  common 
factor,  divide  by  it:  the  last  result  will  be  the  proportional  num 
bers. 

Note. — The  answers  to  the  last,  and  to  all  similar  questions,  will 
be  infinite  in  number,  for  two  reasons: 

1st.  If  the  proportional  numbers  in  column  E  be  multiplied  by 
any  number,  integral  or  fractional,  the  products  will  denote  pro- 
portional parts  of  the  simples. 

2d.  If  the  proportional  numbers  of  any  couplet  be  multiplied  by 
any  number,  the  gain  and  loss  in  that  couplet  will  still  balance, 
and   the   proportional   numbers  in  the  final  result  will  be  changed. 

Rule. — I.  Write  the  prices  or  qualities  of  the  simples  in  a 
column,  hegimmig  with  the  lowest,  and  the  mean  price  or 
quality  at  the  left: 

II.  Opposite  the  first  simple  write  the  part  which  must  he 
taken  to  gain  1  of  the  mean  price,  and  ojyposite  the  other 
simple  of  the  couplet  write  the  part  which  must  he  taken  to 
lose  1   of  the  mean  price,  and  do  the  same  for  each   simjjle : 

III.  When  the  proportional  numhers  are  fractional,  reduce 
them  to  integral  numhers,  and  then  add  those  ivhich  stand 
opposite  the  same  simple  ;  if  the  sums  have  a  common  factor, 
reject  it :    the  result  ivill  denote  the  proportional  parts. 

Examples. 

1.  What  proportions  of  coffee,  at  8  cents,  10  cents,  and  14 
cents  per  pound,  must  be  mixed  together  so  that  the  compoimd 
shall  be  worth  12  cents  per  pound? 

2.  A  merchant  has  teas  worth  40  cents,  65  cents,  and  15 
cent^  a  pound,  from  which  he  wishes  to  make  a  mixture  worth 


ALTERNATE. 


299 


60  cents  a  pound  :    what  is  the  smallest  quantity  of  each  that 
he  can  take  and  express  the  parts  by  whole  numbers  ? 

3.  A  farmer  sold  a  number  of  colts  at  $50  each,  oxen  at 
$40,  cows  at  $25,  calves  at  §10,  and  realized  an  average  price 
of  $30  per  head  :  what  was  the  smallest  number  he  could 
sell  of  each? 

4,  What  is  the  smallest  quantity  of  water  that  must  be 
mixed  with  wine  worth  14s.  and  15s.  a  gallon,  to  form  a 
mixture  worth  13s.  a  gallon,  when  all  the  parts  are  expressed 
by  whole  numbers  ? 

CASE     II. 

332.    When  the  quantity  of  one  of  the  simples  is  given. 
1.    A  farmer  would  mix   rye   worth   80   cents  a  bushel,  and 
corn  worth  15   cents   a  bushel,  with  66  bushels  of  oats  worth 
45  cents  a  bushel,  so  that  the  mixture  shall  be  worth  50  cents 
a  bushel :    how  much  must  be  taken  of  each  sort  ? 


vi't.a.A 

TLVH. 

A. 

B. 

c. 

D. 

E. 

F. 

75-' 

I 

i 

G 

5 

11 

66 

A 

1 

1 

6 

80    ^ 

^0 

1 

1 

6     j 

Analysis. — ^Find  the  proportional  parts  as  in  Case  I.:  tlioy  are 
11,  1  and  1.  But  we  are  to  take  G6  bushels  of  oats  in  the  mix- 
ture; hence,  each  proportional  nmnber  is  to  be  taken  6  times; 
that  is,  as  many  times  as  there  are  units  in  the  quotient  of 
C€  4-  11. 

Rule. — I.  Find  the  proportional  numbers  as  in  Case  L, 
and  vjrite  each  opposite  its  simple: 

II.  Find  the  ratio  of  the  proportional  number  correspond- 
ing to  the  given  simple^  to  the  quantity  of  that  simple  to  be 
takeji,  and  multiply  each  proj^ortional  number  by  it. 

Note. — If  we  multiply  the  numbers  in  cither  or  botlx  of  tlie 
cclauina  C  or   D   \>j   aj^y    number,  the  proportion  of    ili^i   numbers  in 


k 


300 


ALLIGATION". 


column  E  will  be  changed  Thus,  if  we  multiply  column  D  by  12, 
we  shall  have  60  and  12,  and  the  numbers  in  column  E  become 
Q6,  12  and  1,  numbers  which  will  fulfil  the  conditions  of  the  question. 

Examples. 

1.  What  quantity  of  teas  at  12s.  10s.  and  6s.  mast  be 
mixed  with  20  pounds,  at  4s.  a  pound,  to  make  the  mixture 
worth  8s.  a  pound? 

2.  How  many  pounds  of  sugar,  at  *l  cents  and  11  cents  a 
pound,  must  be  mixed  with  t5  pounds,  at  12  cents  a  pound, 
so  that  the  mixture  may  be  worth  10  cents  a  pound? 

3.  How  many  gallons  of  oil,  at  ts.,  Is.  6d.,  and  9s.  a 
gallon,  must  be  mixed  with  24  gallons  of  oil,  at  9s.  6d. 
a  gallon,  so  as  to  form  a  mixture  worth  8s.  a  gallon  ? 

4.  Bought  10  knives  at  $2  each  :  how  many  must  be  bought 
at  Sf  each,  that  the  average  price  of  the  whole  shall  be  $1^? 

5.  A  grocer  mixed  50  lb.  of  sugar  worth  10  cents  a  pound, 
with  sugars  worth  9 J  cents,  tj  cents,  1  cents,  and  5  cents  a 
pound,  and  found  the  mixture  to  be  worth  8  cents  a  pound  : 
how  much  did  he  take  of  each  kind  ? 


CASE      III. 

333.  When  the   quantity   of  the  mixture  is   given. 

1.    A  silversmith  has   four  sorts   of  gold,  viz.,  of  24  carats 

fine,  of  22  carats  fine,  of  20  carats  fine,  and  of  15  carats  fine; 

he  would  make  a  mixture  of  42  ounces  of  11  carats  fine  :  how 

much  must  he  take  of  each  sort  ? 


OPERATION. 


n 


A. 

B. 

C. 

D. 

E 

F. 

a. 

H. 

'15~ 
20-1 

^ 

i 

i 

i 

1 

5 

3 

15 

30 

i 

2 

2 

4 

22   J   1 

i 

2 

2 

4 

.24- 

-J 

} 

2 

2 

4 

rP.OPORTIONAL    TARTS. 

15  +  2  +  2  +  2  =  21  ;     42  -f  21  ==  2. 


ALTERNATE.  301 

Rule. — I.  Find  the  proportional  parts  as  in  Case  I. : 
II.  Divide  the  quantity  of  the  mixture  by  the  sum  of  the 
proportional  parts,  and  the  quotient  will  denote  hoiv  many 
times  each  part  is  to  be  taken.  Multiply  the  parts  separately 
by  this  quotient,  and  each  product  will  denote  the  quantity  of 
the  corresponding  simple. 

Examples. 

1.  A  grocer  has  teas  at  5s.,  6s.,  8s.,  and  9s.  a  pound,  and 
wishes  to  make  a  compound  of  881b.,  worth  7s.  a  pound:  how 
much  of  each  sort  must  be  taken? 

2.  A  liquor  dealer  wishes  to  fill  a  hogshead  with  water, 
and  with  two  kinds  of  brandy  at  $2.50  and  3.00  per  gallon, 
so  that  the  mixture  may  be  worth  $2.25  a  gallon  :  in  what 
proportions  must  he  mix  them  ? 

3.  A  person  sold  a  number  of  sheep,  calves,  and  lambs,  40 
in  all,  for  $48 :  how  many  did  he  sell  of  each,  if  he  received 
for  each  calf  $lf,  each  sheep  $1J,  and  each  lamb  $J? 

4.  A  merchant  sold  20  stoves  for  $180  ;  for  the  largest 
size  he  received  $19  each,  for  the  middle  size,  $7,  and  for  the 
small  size,  $6  :  how  many  did  he  sell  of  each  kind  ? 

5.  A  vintner  has  wines  at  4s.,  6s.,  8s.,  and  10s.  per  gallon  ; 
he  wishes  to  make  a  mixture  of  120  gallons,  worth  5s.  per 
gallon  :  what  quantity  must  he  take  of  each  ? 

6.  A  tailor  has  24  garments,  worth  $144.  He  has  coats, 
pantaloons,   and   vests,    worth   $12,    $5,  and   $2  each,  respect- 

vely  :  how  many  has  he  of  each  ? 

7.  A  merchant  has  4  pieces  of  calico,  each  worth  24,  22, 
20,  and  15  cents  a  yard:  how  much  must  he  cut  from  each 
piece  to  exchange  for  42  yards  of  another  piece,  wortli  17 
cents  a  yard  ? 

8.  A  man  paid  $70  to  3  men  for  35  days'  labor  ;  to  the 
Gi-st  he  paid  $5  a  day,  to  the  second,  $1  a  day,  and  to  the 
third,  $J  a  day  :  how  many  days  did  each  labor  ? 


302  '  CUSTOM-HOUSE  BUSINESS. 


CUSTOM    HOUSE    BUSINESS. 

334.  All  merchandise  imported  into  the  United  States,  must 
be  landed  at  certain  ports,  called  Ports  of  Entry.  On  such 
merchandise  the  General  Government  has  imposed  a  greater  or 
f<!ss  tax,  called  a  duty. 

335.  A  Port  of  Entry  is  a  port  where  foreign  merchandise 
may  be  delivered,  and  where  there  is  a  Custom-house  for  ap- 
praisement and  the  payment  of  duties. 

336.  Tonnage  Duties  are  taxes  levied  on  vessels,  according 
to  their  size,  for  the  privilege  of  entering  ports. 

337.  AH  duties,  levied  by  law,  on  imported  goods,  are  of  two 
kinds  :    Hpecific  and  ad  valorem. 

338.  Specific  Duty  is  a  certain  sum  levied  on  a  particular 
kind  of  goods  named;  as  so  much  per  square  yard  on  cot- 
ton  or   woolen  goods,  so   much  per  ton   weight   on   iron,  &c. 

339.  Ad  Valorem  Duty  is  a  certain  rate  per  cent,  on  the 

invoice, 

340.  An  Invoice  is  an  inventory  of  goods  to  be  landed, 
directed  to  the  person  who  imports  them,  and  stating  their 
cost  at  the  place  from  which  they  were  exported.  Thus,  an 
Q.d  valorem  duty  of  15%  on  English  cloths,  is  a  duty  of  15% 
<m  the  cost  of  the  cloths  imported  from  England. 

341.  The  revenues  of  the  country  are  under  the  general 
direction  of  the  Secretary  of  the  Treasury,  and  to  secure  their 
faithful  collection,  the  government  has  appointed  various  officers 
at  each  port  of  entry,  or  place  where  goods  may  be  landed. 

342.  The  laws  of  Congress  provide,  that  the  cargoes  of  all 
vessels  freighted  with  foreign  goods  or  merchandise,  shall  be 
weighed  or  gauged  by  the  custom-house  officers  at  the  port 
to  which  they  are  consigned.  As  duties  are  only  to  be  paid 
on  the  articles,   and  not  on  the  Jjoxes,  casks,   and  bags   which 


Draft    - 

-    on 

n 

-    from 

<< 

<< 

l( 

(< 

Above  - 

« 

CUSTOM-HOUSE  BUSINESS.  303 

contain  them,  certain  deductions  are  made  from  the  weights  and 
measures,  called  Allowances. 

Gross  Weight  is  the  whole  weight  of  the  goods,  together 
with  that  of  the  hogshead,  barrel,  box,  &c.,  which  contains 
them. 

Net  Weight   is  what  remains  after  all  deductions  arc  made. 
Draft  is  an  allowance  fiom  the   gross  weight  on  account  ol 
waste,  where  there  is  not  actual  tare. 

n>.  lb. 

112  is     1, 

112  to     224      it  is     2, 

224  to     336  "       3, 

336  to  1120  *'       4, 

1120  to  2016         "       7, 

2016  any  weight      "       9, 

consequently,  91b.  is  the  greatest  draft  generally  allowed. 

Tare  is  an  allowance  made,  after  draft  has  been  deducted, 
for  the  weight  of  the  boxes,  barrels,  or  bags  containing  the 
commodity,  and  is  of  three  kinds  :  1st,  Legal  tare,  or  such  as 
is  established  by  law ;  2d,  Customary  tare,  or  such  as  is 
established  by  the  custom  among  merchants  ;  and,  3d,  Actual 
tare,  or  such  as  is  found  by  removing  the  goods  and  actually 
weighing  the  casks  and  boxes  in  which  they  are  contained. 

On  liquors  in  casks,  customary  tare  is  sometimes  allowed  ob 
the  supposition  that  the  cask  is  not  full,  or  what  is  called  its 
actual  wants ;  and  then  an  allowance  of  5  per  cent,  for  leakage. 
A  tare  of  10  per  cent,  is  allowed  on  porter,  ale,  and  beer, 
in  bottles,  on  account  of  breakage,  and  5  per  cent,  on  all  other 
liquors  in  bottles.  At  the  custom-house,  bottles  of  the  com 
mon  size  ar§  estimated  to  contain  2|  gallons  the  dozen.  For 
tables  of  Tare  and  Duty,  see  Ogden  on  the  Tariff  of  1842. 

Examples. 
1.    What  is  the  net   weight   of  25   hogsheads  of  sugar,  the 
gross  weight  being    66  cwt.  3  qr,   141b,;    tare,   111b.  per   hogs- 
head? 


304  CUSTOM-HOUSE  BUSINESS. 

cwt.  qr.    lb, 
66     3     14  gross. 
25  X  11  =  2751b.    -    -      2     3 tare. 

Ans.  64     0     14  net. 


2.  If  the  tare  be   4  lb.   per  hundred,  what  will  be  the  tare 
on  6  T.  2  cwt.  3  qr.  14  lb.  ? 

Tare  for  6  T.  or  120  cwt.  =  480  lb. 
2  cwt.  =      8 
3qr.    =      3 
141b.    =      0^ 
Tare    -    -    491^  lb. 

3.  What  will   be   the   cost   of  3   hogsheads   of  tobacco    at 
89. 4 1  per  cwt.  net,  the  gross  weight  and  tare  being  of 

civt.  qr.   lb.  lb. 


N-o 

..  1     - 

-       9     3     24     - 

-    tare  146 

(< 

2     - 

-     10     2     12     - 

-       "     150 

tt 

3     - 

-     11     1     24     - 

-       "     158 

4.    At  21  cents  per  lb.,  what  will  be  the  cost  of  5  hhd.  of 
coffee,  the  tare  and  gross  weight  being  as  -follows  : 


civf.  qr.   lb. 

lb. 

sTo.   1     -    -     -     6     2     14 

-    .    - 

tare     94 

"     2     -     -     -     9     1     20 

-    _    - 

"     100 

"     8     -     -     -     6     2     22 

-    -    _ 

"       88 

'*    4     -    -    -     T     2     24 

_    .    - 

"       89 

"     5     -    -    -     8     0     13 

.    .    . 

"     100 

at   is   the    net    weight    of 

18  hhd. 

of  tobacco, 

5.  What  is  the  net  weight  of  18  hhd.  of  tobacco,  each 
weighing  gross  8  cwt.  3qr.  14  1b.;  tare  161b.  to  the  cwt.? 

6.  What   is    the  net  weight   and  yalue   of  80   kegs   of  fig?, 
ross  weight  7T.  11  cwt.  3qr.,  tare  121b.  per  cwt.,   at   $2.3 J 

per  cwt.  ? 

7.  A  merchant  bought  19  cwt.  1  qr.  24  lb.  gross  of  tobacco 
in  leaf,  at  $24.28  per  cwt.  ;  and  12  cwt.  3qr.  191b.  gross  in 
rolls,  at  828.56  per  cwt.;  the  tare  of  the  former  was  1491b., 
and  of  the  latter,  491b.  :  what  did  the  tobacco  cost  him,  net? 


CUSTOM-HOUSE  BUSINESS.  305 

8.  A  grocer  bought  17Jlilitl.  of  sugar,  each  lOcwt.  Iqr. 
14  lb.,  draft  7  lb.  per  cwt.,  tare  4  lb.  per  cwt. :  what  is  the 
value   at   $7.50   per  c^vt.  net  ? 

9.  A  merchant  bought  7  hogsheads  of  molasses,  each  weigh- 
ing 4  cwt.  3  qr.  141b.  gross,  draft  7  lb.  per  cwt,  tare  8  lb.  per 
hogshead,  and  damage  in  the  whole  99  j  lb.  :  what  is  the  value 

t  18.45  per  cwt.  net  ? 

10.  The  net  value  of  a  hogshead  of  Barbadoes  sugar  was 
'i^22.50  ;  the  custom  and  fees  812.49,  freight  $5.11,  factorage 
$1.31;  the  gross  weight  was  11  cwt.  Iqr.  151b.,  tare  lljlb. 
per  cwt. :  what  was  the  sugar  rated  at  per  cv/t.  net,  in  the  bill 
of  parcels. 

11.  I  have  imported  87  jars  of  Lucca  oil,  each  containing 
47  gallons ;  what  did  the  freight  come  to  at  $1.19  per  cwt.  net, 
reckoning  lib.  in  11  lb.  for  tare,  and  91b.  of  oil  to  the  gallon? 

12.  A  grocer  bought  5  hhd.  of  sugar,  each  weighing  13  cwt 
Iqr.  121b.,  at  7^  cents  a  pound;  the  draft  was  IJlb.  per 
cwt.,  and  the  tare  5J  per  cent. :  what  was  the  cost  of  the 
net  weight? 

13.  A  wholesale  merchant  receives  450  bags  of  coffee,  each 
weighing  7Clbs.  ;  the  tare  was  eight  per  cent.,  and  the  invoice 
price  lOJ  cents  per  pound.  He  sold  it  at  an  advance  of  33^ 
per  cent.  :  what  was  his  whole  gam,  and  what  his  selling  price  ? 

14.  A  merchant  imported  176  pieces  of  broadcloth,  each 
piece  measuring  46Jyd.,  at  $3.25  a  yard;  what  will  be  the  duty 
at  30  per  cent.? 

15.  What  is  the  duty  on  54  T.  13  cwt.  3  qr.  201b.  of  iron, 
invoiced  at  $45  a  ton,  and  the  duty  33 J  per  cent.  ? 

16.  What  will  be  the  duty  on  225  bags  of  coffee,  each 
weighing  gross  1601b.,  invoiced  at  6  cents  per  pound;  2  per 
cent,  being  the  legal  rate  of  tare,  and  20  per  cent,  the  duty  ? 

17.  What  duty  must  be  paid  on  275  dozen  bottles  of  claret, 
estimated  to  contain  2f  gallons  per  dozen,  5  per  cent,  being 
allowed  for  breakage,  and  the  duty  being  35  cents  per  gallon  f 


306  TONNAGE   OF  VESSELS. 

18.  A  merchant  imports  175  cases  of  indigo,  each  case 
weighing  1961b.  gross  ;  15  per  cent,  is  the  customary  rate  of 
tare,  and  the  duty  5  cents  per  pound ;  what  duty  must  he 
pay  on  the  whole  ? 

19.  What  is  the  tare  and  duty  on  ^5  casks  of  Epsom 
salts,  each  weighing  gross  2  cwt.  2  qr.  24  lb.,  and  invoiced  at 
IJ  cents  per  pound,  the  customary  tare  being  11  per  cent, 
and  the  rate  of  duty  20  per  cent.? 


TONNAGE     OF     VESSELS. 

343.  There  are  certain  custom-house  charges  on  vessels, 
which  are  made  according  to  their  tonnage.  The  tonnage  of 
a  vessel  is  the  number  of  tons  weight  she  will  carry,  and  this 
is  determined  by  measurement. 

[From  the  "Digest,"  by  Andrew  A.  Jones,  of  the  N.  Y.  Custom-house.] 

CuMoni'/iouse  cJiarges  on  all  ships  or  vessels  entering  from  any  foreign 
port  or  place. 

Ships  or  vessels  of  tlie  United  States,  having  three-fourths  of 
tlie  crew  and  all  the  officers  American  citizens,  J9er  ton,         -    $0.06 

Ships  or  vessels  of  nations  entitled  by  treaty  to  enter  at  the 
same  rate  as  American  vessels, .06 

Ships  or  vessels  of  the  United  States  not  having  three-fourths 
of  the  crew  as  above, .50 

On  foreign  ships  or  vessels  other  than  those  entitled  by  treaty,        .50 

Additional  tonnage  on  foreign  vessels,  denominated  light- 
money,  .50 

Licensed  coasters  are  also  liable  once  in  each  year  to  a  duty  of  50 
cents  per  ton,  being  engaged  in  a  trade  from  a  port  in  one  State  to 
a  port  in  another  State,  other  than  an  adjoining  State,  unless  tlie 
officers  and  three-fourths  of  the  crew  are  American  citizens;  to  ascer- 
tain which,  the  crews  are  always  liable  to  an  examination  by  an 
officer. 

A  foreign  vessel  is  not  permitted  to  carry  on  the  coasting  trade; 
but  having  arrived  from  a  foreign   port  with   a  cargo  consigned  to 


TONNAGE   OF  VESSELS.  307 

more  than  one  port  of  the  United  States,  she  may  proceed  coastwise 
with  a  certified  manifest  until  her  voyage  is  completed. 

344.   The  government  estimate  the  tonnage  according  to  one 
I  lie,  while    the   ship-carpenter,  who   builds  the  vessel,  uses  an- 
other. 

G-overiunent  Rule. — I.  Measure,  in  feet,  above  the  ujype 
deck  the  length  of  the  vessel,  from  the  forepart  of  the  main 
stem  to  the  after-part  of  the  stern-post.  Then  measure  the 
breadth  taken  at  the  widest  part  above  the  main  wale  on  the 
outside,  and  the  depth  from,  the  under-side  of  the  deck-plank 
fa  the  ceiling  in  the  hold: 

II.  Fj'om  the  length  take  three-fifths  of  the  breadth,  and 
niultiphj  the  remainder  by  the  breadth  and  depth,  and  the 
product  divided  by  95  will  give  the  tonnage  of  a  si^igle- 
decker ;  and  the  same  for  a  double-decker,  by  merely  making 
the  depth  equal  to  half  the  breadth. 

Cajpenters'  Rule. — Mxdtiply  together  the  length  of  the 
ki'el,  the  breadth  of  the  main  beam,  and  the  depth  of  the 
hold,  and  the  i^roduct  divided  by  95  will  be  the  carpenters^ 
tonnage  for  a  single-decker ;  and  for  a  double-decker,  deduct 
from  tlie  depth  of  the  hold  half  the  distance  between  decks. 

Examples. 

1.  What  is  the  government  tonnage  of  a  single-decker, 
whose  length  is  75  feet,  breadth  20  feet,  and  depth  IT  feet? 

2.  What  is  the  carpenters^  tonnage  of  a  single-decker,  the 
length  of  whose  keel  is  90  feet,  breadth  22  feet  T  inches,  and 
depth  20  feet  6  inches? 

3.  What  is  the  carpenters'  tonnage  of  a  steamship,  double 
decker,  length    154  feet,  breadth  30  feet  8  inches,  and  depth, 

ifter  deducting  aalf  between  decks,  14  feet  8  inches? 

4.  What  is  the  carpenterti'  tonnage  of  a  double-decker,  its 
length  125  feet,  breadth  25  feet  6  inches,  depth  of  hold  34 
feet,  and  distance  between  decks  8  feet? 


308  GENERAL   AVERAGE. 


GENERAL     AVERAGE. 

345.  Average  is  a  term  of  commerce  signifying  a  contribution 
\ty  individuals,  where  the  goods  of  a  particular  merchant  are 
thrown  overboard  in  a  storm,  to  save  the  ship  from  sinking  ; 
or  where  the  masts,  cables,  anchors,  or  other  furniture  of  the 
ship  are  cut  away  or  destroyed,  for  the  preservation  of  the 
vessel.  In  these  and  like  cases,  where  any  sacrifices  are  de- 
liberately made,  or  any  expenses  voluntarily  incurred,  to  pre- 
vent a  total  loss,  such  sacrifice  or  expense  is  the  proper 
subject  of  a  general  contribution,  and  ought  to  be  ratably 
borne  by  the  owners  of  the  ship,  the  freight,  and  the  cargo, 
so  that  the  loss  may  fall  proportionably  on  all.  The  amount 
sacrificed  is  called  the  Jettison. 

343.  Average  is  either  general  or  particular ;  that  is,  it  is 
either  chargeable  to  all  the  interests,  viz.,  the  ship,  the  freight, 
and  the  cargo,  or  only  to  some  of  them.  As  when  losses 
occur  from  ordinary  wear  and  tear,  or  from  the  perils  incident 
to  the  voyage,  without  being  voluntarily  incurred  ;  or  when 
any  particular  sacrifice  is  made  for  the  sake  of  the  ship  only, 
or  the  cargo  only,  these  losses  must  be  borne  by  the  parties 
immediately  interested,  and  are  consequently  defrayed  by  a 
particular  average.  There  are  also  some  small  charges,  called 
petty  or  accustomed  averages,  one-third  of  which  is  usually 
charged  to  the  ship,  and  two-thirds  to  the  cargo. 

No  general  average  ever  takes  place,  except  it  can  be  shown 
that  the  danger  was  imminent,  and  that  the  sacrifice  was 
made  indispensable,  or  was  supposed  to  he  so,  by  the  captain 
and  officers,  for  the  safety  of  the  ship. 

347.  In  different  countries  different  modes  are  adopted  ot 
valuhig  the  articles  which  are  to  constitute  a  general  average. 
In  general,  however,  the  value  of  the  freightage  is  held  to  be 
the  clear  sum  which  the  ship  has  earned  after  seamen's  wages, 
pilotage,  and  all  such  other  charges  as  come  under   the   name 


GENERAL   AVERAGE-  300 

of  petty  charges,  are  deducted  ;   one-third,  and  in  some  cases 
one-half,  being  deducted  for  the  wages  of  the  crew. 

The  goods  lost,  as  well  as  those  saved,  are  valued  at  the 
price  they  would  have  brought,  in  ready  money,  at  the  place 
of  deli  very,  on  the  ship's  arriving  there,  freight,  duties,  and 
all  other  charges  being  deducted  :  indeed,  they  bear  their  pro- 
portions, the  same  as  the  goods  saved.  The  ship  is  valued  at 
the  price  she  would  bring,  on  her  arrival  at  the  port  of  de- 
livery. But  when  the  loss  of  masts,  cables,  and  other  furni- 
ture of  the  sliip  is  compensated  by  general  average,  it  is 
usual,  as  the  new  articles  will  be  of  greater  value  than  the 
old,  to  deduct  one-third,  leaving  two-thirds  only  to  be  charged 
to  the  amount  to  be  contributed. 

Examples. 

1.  The  vessel  Good  Intent,  bound  from  New  York  to  New 
Orleans,  was  lost  on  the  Jersey  beach  the  day  after  sailing. 
She  cut  away  her  cables  and  masts,  and  cast  overboard  a  part 
of  her  cargo,  by  which  another  part  was  injured.  The  ship 
was  finally  got  off,  and  brought  back  to  New  York. 

AMOUNT    OF     LOSS. 

Goods  of  A  cast  overboard,    -        -        -        -    $500 

Damage  of  the  goods  of  B  by  the  jettison,     -      200 
Freight  of  the  goods  cast  overboard,       -        -       100 
Cable,  anchors,  mast,  &c.,  worth  -        8300  ) 
Deduct  one-third,   -        -        -        -  100  ) 

Expenses  of  getting  the  ship  off  the  sands,      -        66 
Pilotage  and  port  duties   going   in  and  out] 
of  the  harbor,  commissions,  &c.,  -       j 

Expenses  in  port,    ------         25 

Adjusting  the  average,     -        -        -    .    -        -  4 

Postage,  ------        - 1 

Total  loss, $1186 


310  GENERAL   AVERAGE. 


ARTICLES   TO    CONTRIBUTE. 

Goods  of  A  cast  overboard, 1500 

Value  of  B's  goods  at  IS".  0.,  deducting  freight,  &c.,         1000 


'*      of  C's       " 

(I               ti 

it 

500 

"      of  D^s       " 

((               tt 

u 

2000 

"      of  E's       " 

it               It 

u 

5000 

Value  of  the  ship, 

- 

- 

- 

2000 

Freight,  after  deducting 

one-third, 

•               • 

800 

$11800 

Then, 

Total  value  : 

total  loss   ;  :   100 

:  per  cent. 

of  loss. 

111800  : 

1180       :  :  100 

:  10; 

hence,  each  loses  10  per  cent,  on  the  value  of  his  interest  in 
the  cargo,  ship,  or  freight.  Therefore,  A  loses  $50  ;  B,  $100 ; 
C,  50  ;  D,  $200  ;  E,  $500  ;  the  owners  of  the  ship,  $280— in 
all,  $1180.  Upon  this  calculation,  the  owners  are  to  lose 
$280  ;  but  they  are  to  receive  their  disbursements  from  the 
contribution :  viz.,  freight  on  goods  thrown  overboard,  $100  ; 
damages  to  ship,  $200  ;  various  disbursements  in  expenses, 
$180  ;  total,  $480  ;  and  deducting  the  amount  of  contribution, 
they  will  actually  receive  $200.     Hence,  the  account  will  stand : 


The  owners  are  to  receive $200 

A  loses  $500,  and  is  to  contribute  $50  ;  hence,  he 

receives     

B  loses  $200,  and   is  to  contribute  $100  ;   hence, 

he  receives 

Total  to  be  received,        -        -  $150 


450 
100 


f  C,  50 

C,  D,   and  E,  have  lost   nothing,  and  are  to  pay  }  D,  200 

t  E,  500 

Total  actually  paid,          -        -        -  $750 


COINS    AND   CURRENCIES.  811 


COINS     AND     CURRENCIES. 

348.  Coins  are  pieces  of  metal,  of  gold,  silver,  or  copper, 
of  fixed  values,  and  impressed  with  a  public  stamp  prescribed 
by  the  country  where  they  are  made.  These  are  called  specie, 
and  are  generally  declared  to  be  a  legal  tender  in  payment  of 
debts.  The  Constitution  of  the  United  States  provides,  that 
the  value  of  gold  and  silver  coins  shall  be  fixed  by  act  of 
Congress. 

The  coins  of  a  country,  and  those  of  foreign  countries  hav- 
mg  a  fixed  value  established  by  law,  together  with  bank-notes 
redeemable  in  specie,  make  up  what  is  called  the  Cun^ency. 

349.  A  Foreign  coin  may  be  said  to  have  four  values  : 

1st.  The  intrinsic  value,  which  is  determined  by  the  amount 
of  pure  metal  which  it  contains : 

2d.   The  Custom-house,  or  legal  value,  which  is  fixed  by  law  : 

3d.  The  mercantile  value,  which  is  the  amount  it  will  sell 
for  in  open  market : 

4th.  The  exchange  value,  which  is  the  value  assigned  to  it 
in  buying  and  selling  bills  of  exchange  between  one  country 
and  another. 

Let  us  take,  as  an  example,  the  English  pound  sterling, 
which  is  represented  by  the  gold  sovereign.  Its  intrinsic  value, 
as  determined  at  the  Mint  in  Philadelphia,  compared  with  our 
gold  eagle,  is  $4,861.  Its  legal  or  custom-house  value  is 
$4.84.  Its  commercial  value,  that  is,  what  it  will  bring  in 
Wall-street,  New  York,  varies  from  14.83  to  $4.86,  seldom 
reaching  either  the  lowest  or  highest  limit.  The  exchange 
value  of  the  English  pound,  is  $4.44^,  and  was  the  legal  value 
before  the  change  in  our  standard.  This  change  raised  the 
cgal  value  of  the  pound  to  $4.84 ;  but  merchants,  and  dealers 
in  exchange,  preferred  to  retain  the  old  value,  which  became 
nominal,  and  to  add  the  diflference  in  the  form  of  a  premium 
on  exchange^  which  is  explained  m  Art.  365.  For  the  values 
of  the  various  coins,  see  Table,  page  406 


312  EXCHANGE. 


EXCHANG-E. 

350.  Exchange  is  a  term  whicli  denotes  the  payment  of 
money  by  a  person  residing  in  one  place  to  a  person  residing 
in  another.  The  payment  is  generally  made  by  means  of  a 
bill  of  exchange. 

351.  A  Bill  of  Exchange  is  an  open  letter  of  request 
from  one  person  to  another,  desiring  the  payment  to  a  third 
party  named  therein,  of  a  certain  sum  of  money  to  be  paid  at 
a  specified  time  and  place.  Of  a  bill  of  exchange  three  copies 
are  made,  and  are  called  a  set  of  exchange.  They  are  sent 
by  different  ways  to  the  drawee,  so  that  in  case  one  is  lost, 
another  may  reach  hhn.  There  are  always  three  parties  to  a 
bill  of  exchange,  and  generally  four  : 

1.  He  who  writes  the  open  letter  of  request,  is  called  the 
drawer  or  maker  of  the  bill  ; 

2.  The  person  to  whom  it  is  directed,  is  called  the  drawee ; 

3.  The  person  to  whom  the  money  is  ordered  to  be  paid  is 
called  the  payee ;   and 

4.  Any  person  who  purchases  a  bill  of  exchange  is  called 
the  h^^yer  or  r^emitter. 

352.  Bills  of  exchange  are  the  proper  money  of  commerce. 
Suppose  Mr.  Isaac  Wilson,  of  the  city  of  New  York,  ships 
1000  bags  of  cotton,  worth  £6000,  to  Samuel  Johns  &  Co., 
of  Liverpool ;  and  at  about  the  same  time  William  James,  of 
New  York,  orders  goods  from  Liverpool,  of  Ambrose  Spooner, 
to  the  amount  of  six  thousand  pounds  sterling.  Now,  Mr. 
Wilson  draws  a  bill  of  exchange  on  Messrs.  Johns  &  Co.,  in 
the  following  form,  viz. : 


Exchange  for  £G000.  New  York,  July  80th,  1846. 

Sixty   days   after   sight    of   this   my   first   Bill   of  Exchange 
(second  and  third  of  the  same  date  and  tenor  unpaid),  pay  to 


EXCUANGK.  313 

David  C.  Jones,  or  order,  six   thousand  pounds  sterling,  with 
or  without  further  notice.  Isaac  Wilsox. 

Messrs.  Samuel  Johna  &  Co.,) 
Merchants,  Liverpool.  ) 

Let  us  now  suppose  that  Mr.  James  purchases  this  bill  of 
David  C.  Jones,  for  the  purpose  of  sending  it  to  Ambrose 
Spooner,  of  Liverpool,  whom  he  owes.  We  shall  then  have 
all  the  parties  to  a  bill  of  exchange  ;  viz.,  Isaac  Wilson,  the 
maker  or  drawer ;  Messrs.  Johns  &  Co.,  the  drawees;  David 
C.  Jones,  the  payee;  and  William  James,  the  buyer  or  re- 
mitter. 

353.  A  bill  of  exchange  is  called  an  inland  bill,  when  the 
drawer  and  drawee  both  reside  in  the  same  country  ;  and  when 
they  reside  in  different  countries,  it  is  called  a  foreign  bill. 
Thus,  all  bills  in  which  the  drawer  and  drawee  reside  m  the 
United  States,  are  inland  bills  ;  but  if  one  of  them  resides  in 
England  or  France,  the  bill  is  a  foreign  bill. 

354.  The  time  at  which  a  bill  is  made  payable  varies,  and 
is  a  matter  of  agreement  between  the  drawer  and  buyer. 
They  may  either  be  drawn  at  si^ht,  or  at  a  certain  number  of 
days  after  sight,  or  at  a  certain  number  of  days  after  date. 

355.  Days  of  Grace  are  a  certain  number  of  days  granted 
to  the  person  who  pays  the  bill,  after  the  time  named  in  the 
bill  has  expired.  In  the  UnHed  States  and  Great  IBritaiu 
three  days  are  allowed. 

356.  In  ascertaining  the  time  when  a  bill,  payable  so  many 
days   after  sight,  or  after  date,  actually  falls  due,  the  day  of 
presentment,  or  the  day  of  the  date,  is  not  reckoned.     When 
the  time  is   expressed  in  months,  calendar  moiUhs  are  alway 
understood. 

If  the  month  in  which  a  bill  falls  due  is  shorter  than  the 
one  in  which  it  is  dated,  it  is  a  rule  not  to  go^on  into  the 
next  month.  Thus,  a  bill  drawn  on  the  28th,  29th,  30th,  or 
31st   of  December,  payable   two   months   after  date,  falls  due 

14 


>14  E,XCHANGE. 


on  Hie   last   of  February,  except  for   the   daj^s   of  grace,  and 
would  be  actually  due  on  the  third  of  March. 


INDORSING    BILLS. 

357.  In  examining  the  bill  of  exchange  drawn  by  Isaac 
Wilson,  it  will  be  seen  that  Messrs.  Johns  &  Co.  are  requested 
to  pay  the  amount  to  David  C.  Jones,  or  order ;  that  is, 
either  to  Jones  or  to  any  other  person  named  by  him.  If  Mr. 
Jones  simply  writes  his  name  on  the  back  of  the  bill,  he  is 
said  to  indorse  it  in  hlanky  and  the  drawees  must  pay  it  to 
any  rightful  owner  who  presents  it.  Such  rightful  owner  is 
called  the  holder,  and  Mr.  Jones  is  called  the  indorser. 

If  Mr.  Jones  writes  on  the  back  of  the  bill,  over  his  signa- 
ture, "Pay  to  the  order  of  William  James,"  this  is  called  a 
special  indorsement,  and  William  James  is  the  indorsee,  and 
he  may  either  indorse  in  blank,  or  write  over  his  signature, 
"  Pay  to  the  order  of  Ambrose  Spooner,"  and  the  drawees, 
Messrs.  Johns  &  Co.,  will  then  be  bound  to  pay  the  amount 
to  Mr.  Spooner. 

A  bill  drawn  payable  to  bearer,  may  be  transferred  by  mere 
delivery. 

ACCEITANCE. 

358.  When  the  bill  drawn  on  Messrs.  Johns  &  Co.  is  pre- 
sented to  them,  they  must  inform  the  holder  whether  or  not 
they  will  pay  it  at  the  expiration  of  the  time  named.  Their 
agreement  to  pay  it  is  signified  by  writing  across  the  face  of 
the  bill,  and  over  their  signature,  the  word  "accepted,"  and 
they  are  then  called  the  acceptors. 

LIABILITIES    OF   THE    PARTIES. 

359.  The  drawee  of  a  bill  does  not  become  responsible  for 
its  payment  until  after  he  has  accepted.  On  the  presentation 
of  the  bill,  if  the  drawee  does  not  accept,  the  holder  should 
immediately  take  means   to   have   the   drawer   and   all   the   in- 


KMCHANQE.  315 

dorsers  notified.  Such  notice  is  called  a  protest,  and  is  giren 
by  a  public  officer  called  a  notary,  or  notary  public.  If  the 
indorsers  are  not  notified  in  a  reasonable  time,  they  are  not 
responsible  for  the  amount  of  the  bill. 

If  the  drawee  accepts  the  bill,  and  fails  to  make  the  pay. 
ment  when  it  becomes  due,  the  parties  must  be  notified  as 
before,  and  this  is  called  protesting  the  hill  for  non-payment. 
If  the  indorsers  are  not  notified  in  a  reasonable  time,  they  are 
not  responsible  for  the  amount  of  the  bill. 

PAR  OF  EXCHANGE — COURSE  OF  EXCHANGE. 

360.  The  intrinsic  par  of  exchange,  is  a  term  used  to  com- 
pare the  coins  of  different  countries  with  each  other,  with  re- 
spect to  their  intrinsic  values  ;  that  is,  with  reference  to  the 
amount  of  pure  metal  in  each.  Thus,  the  English  sovereign, 
which  represents  the  pound  sterling,  is  intrinsically  worth 
$4. 8 CI  in  our  gold,  taken  as  a  standard,  as  determined  at  the 
Mint  in  Philadelphia.  This,  therefore,  is  the  value  at  which 
the  sovereign  should  be  reckoned,  in  estimating  the  par  of  ex- 
change. 

361.  The  commercial  par  of  exchange  is  a  comparison  of 
the  coins  of  different  countries  according  to  their  market  value. 
Thus,  as  the  market  value  of  the  English  sovereign  varies  from 
$4.83  to  $4.85  (Art.  349),  the  commercial  par  of  exchange 
will  fluctuate.  It  is,  however,  always  determined  when  we 
know  the  value  at  which  the  foreign  coin  sells  in  open  market. 

362.  The  course  of  exchange  is  the  variable  price  which  is 
paid  at  one  place  for  bills  of  exchange  drawn  on  another 
The  course  of  exchange   differs   from   the   intrinsic  par   of  ex 

hange,  and  also  from  the  commercial  par,  in  the  same  way 
that  the  market  price  of  an  article  differs  from  its  natural 
price.  The  commercial  par  of  exchange  would  at  all  times  de- 
termine the  course  of  exchange,  if  there  were  no  fluctuations 
in  trade. 


316  EXCHANGE. 

363.  When  the  market  price  of  a  foreign  bill  is  above  the 
commercial  par,  the  exchange  is  said  to  be  at  a  premium,  or 
in  favor  of  the  foreign  place,  because  it  indicates  that  the 
foreign  place  has  sold  more  than  it  has  bought,  and  that 
specie  must  be  shipped  to  make  up  the  difference.  When  the 
market  price  is  below  this  par,  exchange  is  said  to  be  beloiv 
vaVf  or  in  favor  of  the  place  where  the  bill  is  drawn.  Such 
place  will  then  be  a  creditor,  and  the  debt  must  be  paid  in 
specie  or  other  property.  It  should  be  observed,  that  a  favor- 
able state  of  exchange  is  advantageous  to  the  buyer,  but  not 
to  the  seller,  whose  interest,  as  a  dealer  in  exchange,  is  ident'- 
fied  with  that  of  the  place  on  which  the  bill  is  drawn. 

INLAND    BILLS. 

364.  We  have  seen  that  inland  bills  are  those  in  which  the 
drawer  and  drawee  both  reside  in  the  same  country  (Art.  853). 

Examples. 

1.  A  merchant  at  New  Orleans  wishes  to  remit  to  New 
York  $8465,  and  exchange  is  IJ  per  cent,  premium  :  how 
much  must  he  pay  for  such  a  bill  ? 

2.  A  merchant  in  Boston  wishes  to  pay  in  Philadelphia 
$8746.50  ;  exchange  between  Boston  and  Philadelphia  is  IJ 
per  cent,  below  par  :   what  must  he  pay  for  a  bill  ? 

3.  A  merchant  in  Philadelphia  wishes  to  pay  $98*16.40  in 
Baltimore,  and  finds  exchange  to  be  1  per  cent,  below  par : 
what  must  he  pay  for  the  bill  ? 

4.  What  must  be  paid  for  a  draft  of  $10000,  payable  60 
days  after  sight,  on  St.  Louis,  exchange  being  at  a  premium 
of  |"/o,  interest  bemg  charged  at  6%? 

5.  What  amount  of  exchange  on  New  Orleans  can  be 
bought  for  $14815,  the  discount  being  IVo? 

6.  For  what  amount  must  a  bill  of  exchange,  at  30  days,  bo 
drawn,  for  which  I  paid  $9650,  discount  1%,  and  the  interest 
being  6%? 


EXCHANGE.  817 

ENGLAND. 

365.  It  has  been  stated  that  exchanges  between  the 
United  States  and  England  are  made  in  pounds,  shillings  and 
pence,  and  that  the  exchange  value  of  the  pound  sterling  is 
reckoned  at  84.44|-  =  4.4444  +  ;  that  is,  this  value  is  the  base 
in  which  the  bills  of  exchange  are  drawn.  Now  this  value 
*)eing  below  both  the  commercial  and  intrinsic  value,  the  drawers 
of  bills  increase  the  course  of  exchange  so  as  to  make  up 
this  deficiency. 

For  example,  if  we  add  to  the  exchange  value  of  the 
pound,  9  per  cent.,  we  shall  have  its  commercial  value,  very 
nearly. 

Thus,  exchange  value,         -        -        -     =  $4.4444  + 

Nine  per  cent., =      .3999  + 

which  gives,      -         -         -         -  $4.8443 

and  this  is  the  average  of  the  commercial  value,  very  nearly. 
Therefore,  when  the  course  of  exchange  is  at  a  premium  of 
9  per  cent.,  it  is  at  the  commercial  par;  and  as  between 
England  and  this  country,  it  would  stand  near  this  point  but 
for  the  fluctuations  of  trade  and  other  accidental  circumstances. 

Examples. 

1.  A  merchant  in  New  York  wishes  to  remit  to  Liverpool 
£\IQ1  10s.  6d.,  exchange  being  at  8 J  per  cent,  premium: 
how  much  must  he  pay  for  the  bill  in  United  States  money? 

First,  ieiI67  10s.  6d.     -  -  -  =  £1U1.525 

Multiply  by  8J  per  cent.,  -  -  .085 

The  product  is  the  premium  -  -  =         99.239625 

This  product  added,  gives  -  -  iE1266.764625 

which,  reduced  to  dollars  and  cents,  at  the  rate  of  $4.44|  (o 
the  pound,  gives  $5630.008 +,  the  amount  which  must  be  paid 
for  the  bill  in  dollars  and  cents. 

2.  A   merchant   has   to   remit   £36794    8«.  9d.   to   London: 


318  EXCHANGE. 

how  much  must   he   pay  for  a   bill   in   dollars   and  cents,  ex- 
change being  Yf  per  cent,  premium  ? 

3.  A  merchant  in  New  York  wishes  to  remit  to  London 
$67894.25,  exchange  being  at  a  premium  of  9  per  cent.  :  what 
will  be  the  amount  of  his  bill  in  pounds  shillings  and  pence? 

Note. — Add  the  amount  of  tlie  premium  to  tlie  exchange  value  of 
tlie  pound;  viz.,  $4.44^,  which,  in  this  case,  gives  $4.84444;  and  then 
divide  the  amount  in  dollars  by  this  sum,  and  the  quotient  will  be 
the  amount  of  the  bill  in  pounds  and  the  decimals  of  a  pound. 

4.  A  merchant  in  New  York  owes  ^1256  18s.  9d.  in  Lon- 
don ;  exchange  at  a  nominal  premium  of  7  J  per  cent.  :  how 
much  money,  in  United  States  currency,  will  be  necessary  to 
purchase  the  bill? 

5.  I  have  $947.86,  and  wish  to  remit  to  London  £364  18s. 
8d.,  exchange  being  at  S\  per  cent. :  how  much  additional 
money  will  be  necessary? 

6.  Received,  on  consignment  from  London,  an  invoice  of 
English  cloths  amounting  to  £1569  10s.  The  duties  thereon 
amounted  to  |416  ;  storage,  cartage,  and  insurance,  amounted 
to  $85.  The  cloths  were  sold  at  an  advance  of  26  per  cent. 
on  the  invoice.  Supposing  the  commission  2J  per  cent.,  and 
the  premium  of  exchange  12  per  cent.,  what  would  be  the  face 
of  the  bill  of  exchange  that  would  cover  the  net  proceeds? 

FRANCE. 

366.  Accounts  in  France,  and  the  exchange  between  France 
and  other  countries,  are  all  kept  in  francs  and  centimes,  which 
are  hundredths  of  the  franc.  We  see,  from  the  table,  that  the 
value  of  the  franc  is  18.6  cents,  which  gives,  very  nearly,  5 
francs  and  38  centimes  to  the  dollar.  The  rate  of  exchange  is 
computed  on  the  value  18.6  cents,  but  is  often  quoted  by  stat- 
ing the  value  of  the  dollar  in  francs.  Thus,  exchange  on  Paris 
is  said  to  be  5  francs  40  centimes ;  that  is,  one  dollar  will  buy 
a  bill  on  Paris  of  5  francs  and  40  hundredths  of  a  franc. 


EXCHANGE.  310 

Examples. 

1.  A  merchant  ia  New  York  wishes  to  remit  161556  francs 
to  Paris,  exchange  being  at  a  premium  of  1 J  per  cent. :  what 
will  be  the  cost  of  his  bill  in  dollars  and  cents  ? 

Commercial  value  of  the  franc,    -        -         18.6  cents, 
Add  IJ  per  cent.,       -        -        -        .  .279 

Gives  value  for  remitting,    -        -        -        18.819  cents  ; 
then,  161556  X  18.879  =  131632.89724, 

which  is  the  amount  to  be  paid  for  the  bill. 

2.  What  amount,  in  dollars  and  cents,  will  purchase  a  bill 
on  Paris  for  86978  francs,  exchange  being  at  the  rate  of  5 
francs  and  2  centimes  to  the  dollar? 

First,         86978  ~-  5.02  =  $17326.29,  the  amount,  nearly. 
Is  this  bill  above  or  below  par?    What  per  cent.? 

3.  How  much  money  must  be  paid  to  purchase  a  bill  of 
exchange  on  Paris  for  68097  francs,  exchange  being  3  per  cent, 
below  par? 

4.  A  merchant  in  New  York  wishes  to  remit  $16785.25  to 
Paris ;  exchange  gives  6  francs  4  centimes  to  the  dollar  :  how 
much  can  he  remit  in  the  currency  of  Paris? 

HAMBURG. 

367.  Accounts  and  exchanges  with  Hamburg,  are  generally 
made  in  the  marc  banco,  valued,  as  we  see  in  the  table,  at  35 
cents. 

Examples. 

1.  What  amount,  in  dollars  and  cents,  will  purchase  a  bill 
of  exchange  on  Hamburg  for  18649  marcs  banco,  exchange 
being  at  2  per  cent,  premium? 

2.  What  amount  will  purchase  a  bill  for  3678  marcs  banco, 
reckoning  the  exchange  value  of  the  marc  banco  at  34  cents? 
Will  this  be  above  or  below  the  par  of  exchange  ? 


320  EXCHANGE, 


ARBITRATION    OF    EXCHANGE 

368.  Arbitration  of  Exchange  is  the  method  by  which  the 
currency  of  one  country  is  changed  into  that  of  another, 
through  the  medium  of  one  or  more  intervening  currencies, 
with  which  th«  first  and  last  are  compared. 

369.  When  there  is  but  one  intervening  currency,  it  is  called 
Simple  Arbitration ;  and  when  there  is  more  than  one,  it  is 
called  Compound  Arbitration.  The  method  of  performmg  this 
is  called  the  Chain  Rule. 

370.  The  principle  involved  in  arbitration  of  exchange  is 
simply  this :  To  pass  from  one  system  of  values  through 
several  others,  and  find  the  true  proportion  between  the  first 
and  last. 

I.  Let  it  be  required  to  remit  $65t0  to  London,  by  the 
way  of  Paris,  exchange  on  Paris  being  5  francs  15  centimes 
for  $1,  and  the  exchange  from  Paris  to  London  25  francs  and 
80  centimes  for  iEl :  what  will  be  the  value  of  the  remittance 
to  London? 

£1 

Analysis. — $1  =:  5.15  francs :    and  1  franc  = • 

25.80 

If  %\  were  remitted  to  Paris,  it  would  produce  there  5.15  francs; 

an6  if  1  frano  were  remitted  from  Paris  to  London,  it  would  pro' 

duce  there   • 

25.80 

But   $6570  are   remitted  to   Paris;    hence,   they  produce   there 

6570  X  5.15  francs;  and  this  amount  is  remitted  to  London;  hence, 

it  produces  there,   . 

6570  X  5.15  X  -^  =  £1311  9s.  Old. 
25.80 

Hule.—I.  Find  the  value  of  a  single  unit  of  each  of  the 
moneys  named,  in  the  money  of  the  place  next  named : 

II.  Multi2Dly  the  sum  to  be  remitted  by  these  values  in 
succession,  and  the  product  will  be  the  equivalent  in  the 
money  of  the  place  to  ivhich  the  remittance  is  to  be  made. 


EXCHANGE  321 

Examples. 

1.  A  merchant  wishes  to  remit  $4888.40  from  New  York  to 
London,  and  the  exchange  is  at  a  premium  of  10  per  cent. 
He  finds  that  he  can  remit  to  Paris  at  5  francs  15  centimes 
to  the  dollar,  and  to  Hamburg  at  35  cents  per  marc  banco. 
Sow,  the  exchange  between  Paris  and  London  is  25  francs  80 
i-entiines  for  £1  sterling,  and  between  Hamburg  and  London 
13J  marcs  banco  for  £1  sterling  :  how  had  he  better  remit  ? 

OPERATION. 

Ist.  To  London  direct. 
$4888.40  X  j.-sh-i  =  ieiOOO. 


1.03 


2d.   Through  Paris. 


4888.40  X  ^  X       ^     =  j£975.7852  =  £d1o  15s.  8}d. 
1  '^$M 

6.16 


3d.  Through  Hambiirg. 
$4888.40  X  jV  X  T!>?75  =  ^1015.771  =  .£1015  15s.  5d. 

Hence,  the  best  way  to  remit  is  through  Haml3Hrg,  then 
direct ;  and  the  least  advantageous,  through  Paris. 

2.  A  merchant  in  New  York  wishes  to  transmit  $1500  tc 
Vienna,  through  London  and  Hamburg :  what  will  be  tlui 
value  when  received,  if  £1  =  $4.86,  iBl  =  14  marcs  banco,  and 
6  marcs  banco  =  8  florms  ? 

3.  A  merchant  at  Natchez  wishes  to  pay  $10000  in  Boston 
He  transmits  through  New  Orleans  and  New  York.  From 
Natchez  to  New  Orleans  exchange  is  |Vo  premium,  from  New 
Orleans  to  New  York  f/o  discount,  and  from  New  York  to 
Boston  i'/o  discount  :  by  this  exchange,  what  amount  at  NaUoiea 
will  pay  the  debt? 


322  INVOLUTION. 

4.  A,  of  London,  draws  a  bill  of  £862  10s.  ou  B,  of  Cadiz, 
and  remits  the  same  to  C,  of  Havre,  who,  in  turn,  remits  to 
D,  of  Amsterdam,  and  D  remits  to  B,  of  Cadiz  :  how  much 
will  pay  the  bill,  if  1  Spanish  dollar  =  2  florins  15  stivers,  12 
florins  =  26  francs,  and  24  f.  15  c.  =  iEl? 


INVOLUTION. 

371.  A  POWER  OF  A  NUMBER  is  any  product  which  arises 
from  multiplying  the  number  contmually  by  itself. 

The  root,  or  simple  factor,  is  called  the  first  power : 
The  second  power  is  the  product  of  the  root  by  itself: 
The  third  power  is  the  product,  when  the   root   is   taken   3 
times  as  a  factor  : 
The  fourth  power  is  the  product,  when  it  is  taken  4  times  : 
The  fifth  power  is  the  product,  when  it  is  taken  5  times. 

372.  The  number  denoting  how  many  times  the  root  is 
taken  as  a  factor,  is  called  the  exponent  of  the  power.  It  is 
written  a  little  at  the  right  and  over  the  root  :  thus,  if  the 
equal  factor  or  root  is  3, 

3^=    3,  the  1st  power,  root,  or  base. 
32=  3  X  3  =    9,  the  2d   power  of  3. 
33=  3    X  3  X  3  =  27,  the  3d  power  of  3. 
3*  =  3    X  3   X  3  X  3  =  81,  the  fourth  power  of  three. 

373.  Involution  is  the  operation  of  finding  the  powers  ol 
numbers. 

Note. — 1.  There  are  three  things  connected  with  every  power:  1st, 
Tlie  root ;  2d,  The  exponent ;  and  3d,  The  power  or  result  of  the 
multiplication. 

2.  In  finding  any  power,  one  multiplication  gives  the  2d  power: 
hence,  the  number  of  multiplications  is  1  less  Mian  the  exponent. 

Rule. — Multiply  the  number  into  itself  as  many  times  less 
1  as  there  are  units  in  the  exponent,  and  the  last  product 
will  be  the  poioer. 


EVOLUTION. 


32 


Find  the  power 

1.  The  square 

2.  The  square 

3.  The  square 

4.  The  square 

5.  The  square 

6.  The  square 

7.  The  square 

8.  The  square 

9.  The  square 

10.  The  square 

11.  The  square 

12.  The  square 

13.  The  square 

14.  The  square 

15.  The  square 

16.  The  square 

17.  The  square 


Examples, 
of  the  followhi;]^  numbers  : 


of  4? 

18. 

The  cube  of  6? 

of  15? 

19. 

The  cube  of  24  ? 

of  142? 

20. 

The  cube  of  125? 

of  463  ? 

21. 

The  cube  of  136  ? 

of  1340? 

22. 

The  4th  power  of  12  ? 

of  24.6  ? 

23. 

The  5th  power  of  9  ? 

of  .526? 

24. 

The  value  of  (4.25)'  ? 

of  3.125? 

25. 

The  value  of  (1.8)^  ? 

of  .0524? 

26. 

The  value  of  (.45)^^  ? 

off? 

27. 

The  value  of  (}f)^? 

off? 

28. 

The  cube  of  (|)  ? 

of  1  ? 

29. 

The  4th  power  of  f  ? 

off  J? 

30. 

The  value  of  (21)"  ? 

ofifl? 

31. 

The  value  of  {^y  ? 

of  tf? 

32. 

The  value  of  (24f)'? 

of  15x\? 

33. 

The  value  of  (.25)^  ? 

of  225/5  ? 

34. 

The  value  of  (142.5)^? 

EVOLUTION. 

374.  Evolution  is  the  operation  of  finding  the  root  of  a 
number  ;  that  is,  of  finding  one  of  its  equal  factors. 

375.  The  Square  Root  of  a  number  is  tlie  factor  which, 
multiplied  by  itself  once,  will  produce  tlie  number. 

Thus,  8  is  the  square  root  of  64,  because  8  X  8  =  64. 

The  sign  ^  is  called  the  radical  sign.  When  placed  before 
a  number,  it  denotes  that  its  square  root  is  to  be  extracted  : 
Thus,   y'36  =  6.  . 

376.  The  Cube  Root  of  a  number  is  the  factor  whicli,  mul- 
tiplied by  itself  iicnce,  will  produce  the  number. 


824  EXTRACTION    OF   THE    SQUARE    ROOT. 

Thus,  3  is  the  cube  root  of  21,  because  3  x  3  x  P»  =  21. 

We  denote  the  cube  root  by  the  sign  -y/  ,  with  3  written 
over  it :  thus,  -^^27,  denotes  the  cube  root  of  21,  which  is 
equal  to  3.  The  small  figure  3,  placed  over  the  radical,  is 
called  the  index  of  the  root. 

The  terms  Power  and  Root,  are  dependent  on  each  other  : 
thus,  the  power  is  the  product  of  equal  factors  ;  and  the  root 
is  one  of  the  equal  factors. 


EXTRACTION    OF    THE    SQUARE    ROOT. 

377.  The  Square  Root  of  a  number  is  one  of  its  two  equal 
factors.  To  extract  the  square  root  is  to  find  this  factor 
The  first  ten  numbers  and  their  squares  are  : 

1,       2,       3,         4,         5,         6,         1,         8,         9,         10. 
1,       4,       9,       16,       25,       36,       49,       64,       81,       100. 

The  numbers  in  the  first  line  are  the  square  roots  of  those 
in  the  second.  The  numbers  1,  4,  9,  16,  25,  36,  &c.,  haying 
two  exact  equal  factors,  are  called  perfect  squares. 

A  PERFECT  SQUARE  is  a  uumbcr  which  has  two  exact  equal 
factors. 

Note. — The  square  root  of  a  number  less  than  100  %\'lll  be  loss 
than  10 ;  while  the  square  root  of  a  number  greater  than  100  will  be 
greater  than  10:  hence,  the  square  root  of  a  number  expressed  by 
ono  or  two  figures,  is  a  number  expressed  by  one  figure. 

378.     To  find  the  law  of  the  square  of  a  number. 

Any  number  expressed  by  two  or  more  figures  may  be  ro 
garded  as  composed  of  tens  and  units. 

i.   What  is  the  square  of  36  ~  3  tens  -f  6  units? 


OPKRATION. 

3  +  6 
3  +  6 

3 

3'  +  3 

X  6  +  6^ 
X  6 

EXTKACTION   OF  THE    SQUARE  ROOT.  325 

Analysis. — The  square  of  36  is  found 
by  taking  36,  thirty-six  times.  This  is 
done  by  first  taking  it  6  units  times, 
and  then  3  tens  times,  and  adding  the 
products.  36  taken  6  units  times,  gives 
6^  +  3  X  3;    and  taken    3    tens    times,  3'  +  2  (3  X  6)  +  6' 

gives    3  X  6  +  3^;    and    their   sum   is, 
8^  +  2  (3  X  6)  +  6=:  that  is, 

Rule. — The  square  of  a  number  is  equal  to  the  square  of 
the  tens,  plus  twice  the  product  of  the  tens  by  the  units,  plus 
the  square  of  the  units. 

379.    To  find  the  square  root  of  any  number. 

1.  Let  it  now  be  required  to  extract  the  square  root  of 
2025. 

Analysis. — Since  the  number  contains  more  than  two  places  of 
figures,  its  root  "will  contain  tens  and  units.  But  as  the  square  of 
one  ten  is  one  Imndred,  it  follows  that  the  square  of  the  tens  of  the 
required  root  must  be  found  in  the  figures  on  the  left  of  25.  Hence, 
beginning  at  the  right,  we  point  off  the  number  into  periods  of  two 
figures  each. 

We  then  find  the  root  contained  in  20  bun-  operation. 

(h-eds,  which  is  4  tens  or  40.     We  then  square  20  25(45 

4  tens,  which  gives  16  hundred,  and  then  place  jg 

16  under  the  first  period,   and  subtract;   this         85^)42  5 
takes  away  the  square  of  the  tens,  and  leaves  42  5 

425,  w7iich  is  twice  the  product  of  the  tens  by 
the  units  plus  the  square  of  the  units. 

If,  now,  we  double  the  tens,  and  then  divide  the  remainder,  ex- 
clusive of  the  right-hand  figure  (since  that  figure  cannot  enter 
into  the  product  of  the  tens  by  the  units),  by  it,  the  quotient  will 
be  the  units  figure  of  the  root.  If  we  annex  this  figure  to  the  root 
and  to  the  augmented  divisor,  and  then  multiply  the  whole  divisor 
thus  increased  by  it,  the  product  will  be  twice  the  tens  by  the  units, 
plus  the  square  of  the  imits*  and  hence,  we  have  found  both  figures 
of  the  root. 


326  EXTRACTION   OF   THE   SQUARE   KOOT. 

Rule. — I.  Separate  the  given  number  into  periods  of  tivo 
Jlgur'es  each,  by  writing  a  dot  over  the  place  of  units,  a  second 
over  the  place  of  hundreds,  and  so  on  for  each  alternate 
figure  to  the  left: 

II.  Note  the  greatest  square  contained  in  the  period  on  the 
left,  and  place  its  root  on  the  right,  after  the  manner  of  a 
quotient  in  division.  Subtract  the  square  of  this  root  from 
the  first  period,  and  to  the  remainder  bring  down  the  second 
period  for  a  dividend: 

III.  Double  the  root  thus  found  for  a  trial  divisor,  and, 
place  it  on  the  left  of  the  dividend.  Find  how  many  times 
(he  trial  divisor  is  contained  in  the  dividend,  exclusive  of 
its  right-hand  figure,  and  place  the  quotient  in  the  root,  and 
also  annex  it  to  the  divisor: 

TV.  Multiply  the  divisor  thus  increased,  by  the  last  figure 
of  the  root  y  subtract  the  product  from  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  new  divi- 
dend: 

Y.  Doid)le  the  ichole  root  thus  found,  for  a  new  trial  di- 
visor^ and  continue  the  operation  as  before,  until  all  the 
periods  are  brought  down. 

Examples, 

I.   What  is  the  square  root  of  425104  ? 

Analysis. — We.  first   place   a  dot  over  operation. 

the  4,  making  the  right-hand  period  04.  ^2  5i   04(652 

We  then  put  a  dot  over  the   1,  and  also  35 

over  the  2,  making  three  periods.  195'ifi'Sl 

The    greatest    perfect   square  in   42   is  g25 

Sa,  the  root  of  which  is  6.     Placing  6  in  ^     ,^ 

the   root,  subtracting   its   square  from   42,  ^   ""  ofOi 

^d   bringing    down   the   next   period  51,  

we  have  651  for  a  dividend;  and  by  doubling  the  root,  we  liave  12 
for  a  trial  divisor.  Now,  13  is  contained  in  65,  5  times.  Place  5 
both  in  the  root  and  in  the  divisor;  then  multiply  125  by  5;  sub- 
tract the  product,  and  bring  down  the  next  period. 


EXTKACTION    OF   THE   SQUARE   ROOT. 


327 


We  must  now  double  the  whole  root  G5  for  a  new  trial  divisor; 
or  we  may  take  the  first  divisor,  after  having  doubled  the  last  figure 
6;  then  dividing,  we  obtain  2,  the  third  figure  of  the  root. 

Notes. — 1.  The  left-hand  peri(xl  may  contain  but  one  figure;  each  of 
the  others  will  contain  two. 

2.  If  any  trial  divisor  is  greater  than  its  dividend,  the  correspond- 
ing root  figure  will  be  a  cipher. 

3.  If  the  product  of  the  divisor  by  any  figure  of  the  root  exceed 
the  corresponding  dividend,  the  root  figure  is  too  large,  and  must  be 
diminished. 

4.  There  will  be  as  many  figures  in  the  root  as  there  are  i)eriod8 
in  the  given  number. 

5.  If  the  given  number  is  not  a  perfect  square,  there  will  be  a 
remainder  after  all  the  periods  are  brought  down.  In  this  case, 
lx3riods  of  ciphers  may  be  annexed,  forming  new  periods,  each  of 
which  will  give  one  decimal  place  in  the  root. 

2.    What  is  the  square  root  of  758692  ? 

OPERATION. 


75  86  92(871.029  +. 
64 


Analysis.  —  After  using  all 
the  periods  of  the  given  num- 
ber, we  annex  periods  of  deci- 
mal ciphers,  each  of  which  gives 
one  decimal  place  in  the  quo- 
tient. 


What  are  the  square  roots  of 

3.  Square  root  of  49? 

4.  Square  root  of  144? 

5.  Square  root  of  225  ? 

6.  Square  root  of  2304  ? 

7.  Square  root  of  7994? 

8.  Square  root  of  6275025  ? 


167)11  86 
11   69 


1741)17  92 
17  41 


174202)510000 
348404 


1742049)16159600 
15678441 

48ll59  Rem. 

the  following  numbers  : 

9.  -v/10000  =  what  No.  ? 


10.  ^2768456  =  what  No.  ? 

11.  i/3«T54  =:  what  No.  ? 

12.  VH>00000  =  what  No.? 

13.  -/y 6Y2^  =  what  No.?. 

14.  -v/30225''  =  what  No.  ? 


328 


EXTRACTION   OF   THE   SQUARE   ROOT. 


380.    To  extract  the  square  root  of  a  fraction. 

1.   What  is  the  square  root  of  .6? 
Analysis. — "We  first  annex  one  cipher,  operation. 

to  make  even  decimal  places;  for,  one  .60(.*It4  + 

decimal  multiplied  by  itself  will  give  49 

wo  places  in  the  product.  We  then 
extract  the  root  of  the  first  period, 
and  to  the  remainder  annex  a  decimal 
period;  and  so  on,  till  we  have  found 
a  sufficient  number  of  decimal  places. 


2.  What  is  the  square  root  of  if' 

Analysis. — The  square  root  of  a 
fraction  is  equal  to  the  square  root  of 
the  numerator  divided  by  the  square 
root  of  the  denominator. 

3.  What  is  the  square  root  of  f  ? 

Analysis. — When  the  terms  are  not 
perfect  squares,  reduce  the  common 
fraction  to  a  decimal,  and  then  extract 
the  square  root  of  the  decimal. 


141)1100 
1029 
1544)7100 
6176 
924  rem. 


OPERATION. 

—  -s/Ie  —  4 

~    -^25  5' 


OPERATION. 

J  -  .75  ; 

-/!=  V'?15  =  .8545  + 


Rule. — I.  If  the  fraction  is  a  decimal,  point  off  the 
periods  from  the  decimal  point  to  the  right,  annexing  ciphers 
if  necessary,  so  that  each  period  shall  contain  two  places, 
and  then  extract  the  root  as  in  integral  numbers: 

II.  If  the  fraction  is  a  common  fraction,  and  its  terms 
verfect  squares,  extract  the  square  root  of  the  numerator  and 
denominator  separately : 

III.  If,  after  being  reduced  to  their  lowest  terms,  the 
numerator  and  denominator  are  not  perfect  squares,  reduce 
the  fraction  to  a  decimal,  and  then  extract  the  square  root 
of  the  result. 


EXTRACTION   OF  THE  SQUARE  ROOT. 


329 


Examples. 
What  are  the  square  roots  of  the  following  numbers? 


4.  Square  root  of  f  f  ? 

5.  Square  root  of  ^^^^  ? 

6.  Square  root  of  .0196? 

7.  Square  root  of  6.25  ? 

8.  Square  root  of  278.89  ? 

9.  Square  root  of  .205209  ? 

10.  Square  root  of  J? 

11.  Square  root  of  |J  ? 

12.  Square  root  of  J^  ? 

13.  Square  root  of  5 J-  ? 

14.  Square  root  of  .7994? 

15.  Value  of  -yJ^Mi  ? 


16.  Square  root  of  .60794? 

17.  Value  of  -v/.02220i  ? 

18.  Value  of  -^25.1001  ? 

19.  Value  of  -v/l9G.425  ? 

20.  Value  of  yTs"? 

21.  Value  of  n/lfls  ? 

''    6  2  4  1    * 

22.  Value  of  >/T  ? 

23.  Value  of  ^X  ? 

24.  Value  of  ^l35  ? 

25.  Value  of  -v/."784  ? 

26.  Square  root  of  5647.5225? 

27.  Square  root  of  160048.0036? 


Applications  in  Square  Root. 

381.  A  TRIANGLE  Is  a  plain  figure  which  has  three  sides  and 
three  angles. 

If  a  straight  line  meets  another  straight  line, 
making  the  adjacent  angles  equal,  each  is  called  a 
right  angle  ;  and  the  lines  are  said  to  be  perpen- 
dicular to  each  other. 

382.  A  RIGHT-ANGLED  triangle  is  one  which 
has    one   right    angle.      In    the    right-angled 
triangle   ABC,    the    side    AC,    opposite    the 
right   angle   B,    is   called    the   hypothenuse ;      "^^ 
the   side  AB,  the   base;    and    the    side    BC,  a 
the  pei^endicular. 


Base. 


383.   A  SQUARE  is  a  figure   bounded  by.  four  equal   sides,  at 
rij^ht  an"cles  to  each  other. 


384.    In  a  right-angled  triangle  the   square  described  on  the 


330 


EXTRACTION    OF  THE   SQUARE  HOOT.' 


D 

_. 

hypothenuse  is  equal  to  the  sum  of  the  squares  described  on 
the  other  two  sides. 

Thus,  if  ACB  be  a  right-angled 
triangle,  right-angled  at  C,  then 
will  the  large  square,  D,  described 
in  the  hypothenuse  AB,  be  equal 
to  the  sum  of  the  squares  F  and 
E,  described  on  the  sides  AC  and 
CB.  This  is  called  the  carpenter's 
theorem.  By  counting  the  small 
squares  in  the  large  square  D, 
you  will  find  their  number  equal 
to  that  contained  in  the  small 
squares  F  and  E.     In  this  triangle 

the  hypothenuse  AB  =  5,  AC  =  4,  and  CB  =  3.  Any  num- 
bers having  the  same  ratio,  as  5,  4,  and  3,  such  as  10,  8,  and 
6  ;  20,  16,  and  12,  &c.,  will  represent  the  sides  of  a  right- 
angled  triangle. 

385.  When  the  base  and  perpendicular  are  known,  to  find  the 
hypothenuse. 

Analysis. — Wishing  to   know  the  distance  from  0 

A  to  the  top  of  a  tower,  I  measured  the  height 
of  the  tower,  and  found  it  to  bo  40  feet ;  also  the 
distance  from  A  to  B,  and  found  it  30  feet:  what 
was  the  distance  from  A  to  02 

AB  =  30;  AB2  =  30^  =    900 

BO  =  40;  BO-  =  40=  =  1600 

AC2=AB2-f  B02  =  900  +  1600 
AC  =  v/2500  =  50  feet. 

Rule. — Square  the  base  and  square  the  perpendicular,  add 
the  results,  and  then  extract  the  square  root  of  their  sum. 

386.  To  find   one   side,  when  we   know  the  hypothenuse  and 
the  other  side. 

1.  The  length  of  a  ladder  wliich  will  reach  from  the  middle 


EXTRACTION   OF  THE   SQUARE  ROOT.  331 

of  a  street  80  feet  wide  to  the  eaves  of  a  house,  is  50  feet : 
what  is  the  height  of  the  house? 

Analysis. — Since  the  square  of  the  length  of  the  ladder  is  equal 
to  the  sum  of  the  squares  of  half  the  width  of  the  street  and  the 
height  of  the  house,  the  square  of  the  length  of  tlie  ladder  dimin- 
ished by  the  square  of  half  the  width  of  tlie  street,  will  he  equa 
to  the  square  of  the  height  of  the  house :   hence. 

Rule. — Square  the  hypothenuse  and  the  known  side,  and 
take  the  difference;  the  square  root  of  the  difference  will  be 
the  other  side. 

Examples. 

1.  A  general  having  an  army  of  111649  men,  wished  to 
form  them  into  a  square  :  how  many  should  he  place  on  each 
front  ? 

2.  In  a  square  piece  of  pavement  there  are  48841  stones, 
of  equal  size,  one  foot  square  :  what  is  the  length  of  one  side 
of  the  pavement? 

3.  In  the  center  of  a  square  garden,  there  is  an  artificial 
cu-cular  pond,  covering  an  area  of  810  square  feet,  which  is  -|^ 
of  the  whole  garden  :  how  many  rods  of  fence  will  inclose  the 
garden  ? 

4.  Let  it  be  required  to  lay  out  6t  A.  2  R.  of  land  in  the 
form  of  a  rectangle,  the  longer  side  of  which  is  to  be  three 
times  as  great  as  the  less  :  what  is  its  length  and  width  ? 

5.  A  farmer  wishes  to  set  out  an  orchard  of  8200  dwarf 
pear-trees.  He  has  a  field  twice  as  long  as  it  is  wide,  which 
he  appropriates  to  this  purpose.  He  sets  the  trees  12  feet 
apart,  and  in  rows  that  are  Ukewise  12  feet  apart :  how 
many  rows  will  there  be,  how  many  trees  in  a  row,  and  how 
much  land  will  they  occupy  ? 

G.  There  is  a  wall  45  feet  high,  built  upon  the  bank  of  a 
-tream  60  feet  wide:  how  long  must  a  ladder  be  that  will 
reach  from  the  oae  €ide  of  the  stream  to  the  top  of  the  wall 
on  the  other? 


632  EXTRACTION   OF  THE   SQUARE   ROOT. 

7.  A  boy  having  lodged  his  kite  in  the  top  of  a  tree,  finds 
that  by  letting  out  the  whole  length  of  his  line,  which  he 
knows  to  be  225  feet,  it  will  reach  the  ground  180  feet  from 
the  foot  of  the  tree  :  what  is  the  height  of  the  tree  ? 

8.  There  are  two  buildings  standing  on  opposite  sides  of  the 
street,  one  39  feet,  and  the  other  49  feet  from  the  ground  to 
the  eaves.  The  foot  of  a  ladder  65  feet  long  rests  upon  the 
ground  at  a  point  between  them,  from  which  it  will  touch  the 
eaves  of  either  building  :  what  is  the  width  of  the  street? 

9.  A  tree  120  feet  high  was  broken  off  in  a  storm,  the  top 
striking  40  feet  from  the  roots,  and  the  broken  end  resting 
upon  the  stump  :  allowing  the  ground  to  be  a  horizontal  plane, 
what  was  the  height  of  the  part  standing? 

10.  What  will  be  the  distance  from  corner  to  corner,  through 
the  center  of  a  cube,  whose  dimensions  are  5  feet  on  a  side? 

11.  Two  vessels  start  from  the  same  point,  one  sails  due 
north  at  the  rate  of  10  miles  an  hour,  the  other  due  west  at 
the  rate  of  14  miles  an  hour  :  how  far  apart  will  they  be  at 
the  end  of  2  days,  supposing  the  surface  of  the  earth  to  be  a 
plane  ? 

12.  How  much  more  will  it  cost  to  fence  10  acres  of  land, 
in  the  form  of  a  rectangle,  the  length  of  which  is  four  times 
its  breadth,  than  if  it  were  in  the  form  of  a  square,  the  cost 
of  the  fence  being  12.50  a  rod? 

13.  What  is  the  diameter  of  a  cylindrical  reservoir  contain- 
ing 9  times  as  much  water  as  one  25  feet  in  diameter,  the 
height  being  the  same  ? 

Note. — If  two  volumes  have  the  same  altitude,  their  contents  will 
be  to  each  other  in  the  same  proportion  as  their  bases;  and  if  the 
bases  are  similar  figures  (that  is,  of  like  form),  they  will  be  to  each 
other  as  the  squares  of  their  diameters,  or  other  like  dimensions. 

14.  If  a  cylindrical  cistern  eight  feet  in  diameter  will  hold 
120  barrels,  what  must  be  the  diameter  of  a  cistern  of  the 
same  depth  to  hold  1500  barrels  ? 


CUBF.    ROOT.  333 

15.  If  a  pipe  3  inches  in  diameter  will  discharge  400  gallons 
in  3  minutes,  what  must  be  the  diameter  of  a  pipe  that  will 
discharge  IGOO  gallons  in  the  same  time? 

16.  What  length  of  rope  must  be  attached  to  a  halter  4 
feet  long,  that  a  horse  may  feed  over  2 J  acres  of  ground? 

17.  Three  men  bought  a  grindstone,  which  was  4  feet  in 
liameter  :  how  much  of  the  radius  must  each  grind  off  to  usd 
up  his  share  of  the  stone? 

CUBE     ROOT. 

387.  The  Cube  Root  of  a  number  is  one  of  its  three  equal 
factors. 

Thus,  2  is  the  cube  root  of  8  ;  for,  2  x  2  x  2  =  8  :  and  3 
is  the  cube  root  of  2t  ;  for,  3x3x3  =  27. 

To  extract  the  cube  root  of  a  number,  is  to  find  one  of 
its    three  equal  factors. 

1,         2,         3,         4,         6,         6,         7,         8,         9, 
1  8         27        64       125      216      343      512      729 

The  numbers  in  the  first  line  are  the  cube  roots  of  the  cor- 
responding numbers  of  the  second.  The  numbers  of  the  second 
line  are  called  perfect  cubes. 

,A  Perfect  Cube  is  a  number  which  has   three  exact  equal 
factors.     By  examining  the  numbers  in  the  two  lines,  we  see, 

1st.  That  the  cube  of  units  cannot  give  a  higher  order  than 
hundreds : 

2d.  That  since  the  cube  of  one  ten  (10)  is  1000,  and  the 
cube  of  9  tens  (90),  729,000,  the  cube  of  tens  wilt  not  give  a 
lower  denomination  than  thousands,  nor  a  higher  denomina- 
tion than  hundreds  of  thousands. 

Hence,  if  a  number  contains  more  than  three  figures,  its 
cube  root  will  contain  more  than  one  ;  if  it  contains  more  than 
six,  its  root  will  contain  more  than  two,  and  so  on ;  every 
additional  three  figures  giving  one  additional  figure  in  the  root, 


U-. 

=  10  + 

6 

10  + 

6 

60  + 

36 

100  + 

60 

100  + 

120  + 

36 

10  + 

6 

83-1  CUBE    MOOT. 

and  the  figures  whicli  remain  at  the  left  hand,  although  less 
than  three,  will  also  give  a  figure  in  the  root.  This  law  ex- 
plains the  reason  for  pointing  off  into  periods  of  three  figures 
each. 

388.   Let    us    see   how  the   cube   of  any  number,  as    16,  is 
^ormed.     Sixteen  is  composed  of  1   ten  and   6  units,  and  may 
le   written,    10  +  6.     To   find   the   cube   of   16  =  10  +  6,  we 
must  multiply  the  number  by  itself  twice. 
To  do  this  we  place  the  number  thus. 

Product  by  the  units,    -        -        -        - 
Product  by  the  tens,     -        -        -        - 

Square  of  16, 

Multiply  again  by  16,  - 

Product  by  the  units,   -        -        -        -         600  +    720  +  216 

Product  by  the  tens,     -        -         -  1000  +  1200  +    360 

Cube  of  16,       -         -         -         -  1000  +  1800  +  1080  +  216 

1.  By  examining  the  parts  of  this    number,  it  is   seen  that 
the  first  part  1000  is  the  cube  of  the  tens;  that  is, 

10  X  10  X  10  =  1000  : 

2.  The   second  part   1800   is  three  times  the  square  of  the 
tens  multiplied  by  the  units;  that  is, 

3  X  (10)=  X6  =  3xl00x6=  1800 : 

3.  The   third   part   1080   is   three  times  the  square  of  the 
units  multiplied  by  the  tens;  that  is, 

3  X  6'^  X  10  =  3  X  36  X  10  =  1080  : 

4.  The  fourth  part  is  the  cube  of  the  units;   that  is, 

6»  =  6  X  6  X  6=  216. 

1.  What  is  the  cube  root  of  the  number  4096  ? 

Analysis. — Since    the    num-  operation. 

ber  contains  more   than  three  ;  aoA/i^ 

-  ,  -        ,  4  uyD(lo 

figures,  we  know  that  the  root  j 

will  contain  at  least  units  and 

tens. 


12  X  3  =  3)3  0     (9-8-t-6 


Separating    tlie  three  right-  ^^    —  4  096 


ci'ni.:  HOOT.  335 

hand  figures  from  tlie  4,  we  know  that  the  cube  of  the  tens  will 
be  found  in  the  4;    and  1  is  the  greatest  cube  in  4. 

Hence,  wc  place  the  root  1  on  the  right,  and  this  is  the  tens 
of  the  required  root.  We  then  cube  1,  and  subtract  the  result 
from  4,  and  to  the  remainder  we  bring  down  the  first  figure  0 
of  the  next  period. 

We  have  seen  that  the  second  part  of  the  cube  of  16,  viz., 
^800,  is  three  times  the  square  of  the  tens  multiplied  by  the  units; 
and  hence,  it  can  have  no  significant  figure  of  a  less  denomination 
than  hundreds.  It  must,  therefore,  make  up  a  part  of  the  30  liun- 
dreds  above.  But  this  30  hundreds  also  contains  all  the  hundreds 
which  come  from  the  3d  and  4th  parts  of  the  cube  of  16.  If  it 
were  not  so,  the  30  hundreds,  divided  by  three  times  the  square 
of  the  tons,  would  give  the  unit  figure  exactly. 

Forming  a  divisor  of  three  times  the  square  of  the  tens,  we 
find  the  quotient  to  be  ten;  but  this  we  know  to  be  too  large. 
Placing  9  in  the  root,  and  cubing  19,  we  find  the  result  to  be 
0859.  Then  trying  8,  we  find  the  cube  of  18  still  too  large;  but 
when  we  take  6,  we  find  the  exact  number.  Hence,  tlie  cube  root 
of  4096  is  16. 

389.    Hence,  to  find  the  cube  root  of  a  number: 

Rule. — I.  Separate  the  given  mimher  into  periods  of 
three  figures  each^  beginning  at  the  rights  by  placing  a  dot 
over  the  place  ofunits^  a  second  over  the  place  of  thousands^ 
and  so  on  over  each  third  figure  to  the  left:  the  left-hand 
period  icill  often  contain  less  than  three  places  of  figures : 

II.  Note  the  greatest  perfect  cube  in  the  first  period^  a?id 
set  its  root  on  the  rights  aftei'  the  manner  of  a  quotient  i7i 
division.  Subtract  the  cube  of  this  nvmber  from  the  first 
2:>eriod,  and  to  the  remainder  bring  down  the  first  figure  of 
tiie  next  period  for  a  dividend: 

III.  Take  three  times  the  square  of  the  root  just  found 
for  a  t)  ial  divisor,  and  see  how  often  it  is  contained  in  the 
dividend^  and  place  the  quotient  for  a  second  figure  of  the 
root.  Then  cube  the  figures  of  the  root  thus  founds  and 
if  tlieir  cube  b<    greater  than   the  first    two  x>eriods  of  th^ 


336  CUBE    ROOT. 

given  number ^  diminish  the  last  figure  j  hut  if  it  be  less^ 
subtract  it  from  the  first  two  periods^  and  to  the  remainder 
bring  down  the  first  figure  of  the  next  'period  for  a  new 
dividend: 

IV.  Take  three  times  the  square  of  the  whole  root  for  a 
second  trial  divisor,  and  find  a  third  figure  of  the  root  as 
before.  Cube  the  whole  root  thus  found,  and  subtract  the 
result  from  tJie  first  three  periods  of  the  given  number  loheu 
it  is  less  than  that  number  ^  but  if  it  is  greater ,  diminish 
the  last  figure  of  the  root:  proceed  in  a  similar  way  for 
all  the  periods. 

Examples. 

I.   What  is  the  cube  root  of  20t968t5? 

OPERATION. 

20  196  815(215 
2^=     8 

2='  X  3  =  12)121 

Fh-st  two  periods,     -    -    -    -    20  196 
(2iy  =  21  X  21  X  21  =  19  683 

3  X  (2iy  =  2181)li  138 

First  three  periods,  ...    -     20  196  815 
(215)^=  215  X  215  X  215  =  20  196  815 

Find  the  cube  roots  of  the  followmff  numbers  : 


1.  Cube  root  of  1128? 

2.  Cube  root  of  111649? 

3.  Cube  root  of  46656  ? 

4.  Cube  root  of  15069223? 


6.  Cube  root  of  5135339  ? 
6.  Cube  root  of  48228544  ? 
1.  Cube  root  of  84604519? 
8.  Cube  root  of  28991029248  ? 


390.    To  extract  the  cube  root  of  a  decimal  fraction. 

Rule. — Annex  ciphers  to  the  decimal,  if  necessary,  so  that 
it  shall  consist  of  3,  6,  9,  &c.,  decimal  places.  Then  put  the 
first  point  over  the  place  of  thousandths,  the  second  over  the 


CUBE    ROOT. 


337 


place    of  millionths^  and  so  on    over  every  third   place    to 
the  right;  after  which,  extract  the  root  as  in  whole  numbers. 

Notes. — 1.  There  "will  be  as  many  decimal  places  in  the  root  as 
there  are  jxjriods  of  decimals  in  the  given  number. 

2.  If,  in  extracting  the  root  of  a  number,  there  is  a  remainder  after 
all  the  periods  have  been  brought  down,  periods  of  ciphers  may  be 
annexed  by  considering  them  as  decimals. 


Examples. 
Fnid  the  cube  roots  of  the  followin":  numbers  : 


1.  Cube  root  of  8.343  ? 

2.  Cube  root  of  1T28.729  ? 

3.  Cube  root  of  .0125  ? 

4.  Cube  root  of  19683.46656? 


5.  Cube  root  of  .38T420489? 

6.  Cube  root  of  .000003375  ? 
T.  Cube  root  of  .0066592? 

8.  Value  of  -^81.729? 


391.    To  extract  the  cube  root  of  a  common  fraction. 

Rule. — I.  Heduce  compound  fractions  to  simple  09ies, 
mixed  numbers  to  improper  fractions^  and  then  reduce  the 
fraction  to  its  loicest  terms: 

II.  Extract  the  cube  root  of  the  numerator  and  denom- 
inator separately^  if  t/iey  have  exact  roots ;  but  if  either  of 
them  has  not  an  exact  root,  reduce  the  fraction  to  a 
deciwMl,  and  extract  the  root  as  in  the  last  case. 


Examples. 
Find  the  cube  roots  of  the  following  fractions 


1.  Cube  root  of  ^*^  ? 

2.  Cube  root  of  ^  ? 

3.  Cube  root  of  31^? 

4.  Cube  root  of  91 J  ? 

5.  Cube  root  of  ffj  ? 


6.  Cube  root  of  yJlf^  ? 

7.  Cube  root  of  ^HjW  ? 

8.  Cube  root  of  i|ff|  ? 

9.  Cube  root  of  7f  ? 
10.   Cube  root  of  66f  ? 


15 


338  Cl^BE   ROOT. 

Applications. 

1.  What  must  be  the  dimensions  of  a  cubical  bin,  that  its 
volume  of  capacity  may  be  19683  feet? 

2.  If  a  cubical  body  contains  6859  cubic  feet,  what  is  the 
length  of  one  side  ?  what  the  area  of  its  surface  ? 

3.  The  volume  of  a  globe  is  46656  cubic  inches  ;  what 
would  be  the  side  of  a  cube  of  equal  solidity  ? 

4.  A  person  wishes  to  make  a  cubical  cistern,  which  ^hall 
hold  150  barrels  of  water:  what  must  be  its  depth? 

5.  A  farmer  constructed  a  bin  that  would  contain  1500 
bushels  of  grain  ;  its  length  and  breadth  were  equal,  and  each 
half  the  height  :  what  were  its  dimensions  ? 

6.  What  is  the  difference  between  half  a  cubic  yard,  and 
a  <!ube  whose  edge  is  half  a  yard? 

I.  A  merchant  paid  $911.25  for  some  pieces  of  muslin.  He 
paid  as  many  cents  a  yard  as  there  were  yards  in  each  piece, 
and  there  were  as  many  pieces  as  there  were  yards  in  one 
piece  :  how  many  yards  were  there,  and  how  much  did  he  pay 
a  yard? 

8.  If  a  sphere  3  feet  in  diameter  contains  14.13^2  cubic 
feet,  what  are  the  contents  of  a  sphere  6  feet  in  diameter? 

3'    :    6'    :  :    14.1372    :    113.0976.  Ans. 

9.  If  a  ball  2 J  inches  in  diameter  weighs  8  pounds,  bow 
much  will  one  of  the  same  kind  weigh,  that  is  5  inches  in 
diameter? 

10.  What  must  be  the  size  of  a  cubical  bin,  that  will  con- 
ain  8  times  as  much  as  one  that  is  4  feet  on  a  side  ? 

II.  How  many  globes,  6  inches  in  diameter,  would  be  re 
quired  to  make  one  12  inches  in  diameter? 

12.  If  a  ball  of  silver,  one  unit  in  diameter,  is  v^orth  ^8, 
what  will  be  the  value  of  one  5^  units  in  diameter? 

IS.    If  a  plate  of  silver,  6  inches   long,  8  inches  wide^  and 


ARITHMETICAL    PROGRESSION.  339 

}  inch  thick,  is  worth  $100,  what  will  be  the  dimensions  of  a 
similar  plate,  of  the  same  metal,  worth  $800? 

14.  If  a  man  can  dig  a  cellar  12  feet  long,  10  feet  wide, 
and  4J  feet  deep,  in  3  days,  what  will  be  the  dimensions  of  a 
similar  cellar,  requiring  24  days  to  dig  it,  working  at  the  same 
rate,  and  the  ground  being  of  the  same  degree  of  hardness? 

15.  If  I  put  2  tons  of  hay  in  a  stack  10  feet  high,  ho^ 
high  must  a  similar  stack  be  to  contain  16  tons? 

IG.  Four  women  bought  a  ball  of  yarn  6  inches  in  diameter, 
and  agreed  that  each  should  take  her  share  separately  from 
the  outer  part  of  the  ball :  how  much  of  the  diameter  did 
each  wind  oflf? 


ARITHMETICAL     PROGRESSION. 

392.  An  Arithmetical  Progression  is  a  series  of  numbers 
in  which  each  is  derived  from  the  one  preceding,  by  the  ad- 
dition or  subtraction  of  the  same  number. 

The  common  difference  is  the  number  which  is  added  or 
subtracted. 

393.  When  the  series  is  formed  by  the  continued  addition 
of  the  common  difference,  it  is  called  an  increasing  series  ; 
and  when  it  is  formed  by  the  subtraction  of  the  common  dif- 
ference, it  is  called  a  decreasing  series  :   thus, 

2,     5,     8,    11,    14,    It,    20,    23,        is  an  increasing  series 
23,   20,    VI,   14,    11,     8,     5,     2,        is  a  decreasing  series. 

The  several  numbers  are  called  terms  of  the  progression 
The  first  and  last  terms  are  called  the  extremes,  and  the  in 
vcrmediate  terms  are  called  the  means. 

394.  In  every  arithmetical  progression  there  are  five  parts, 
any  three  of  which  being  given  or  known,  the  remaining  two 
can  be  determined.    They  are, 


340  ARITHMETICAL   PROGRESSION. 

1st,  The  first  term  ; 

2d,  The  last  term  ; 

3d,  The  common  difference  ; 

4th,  The  number  of  terms  ; 

5  th,  The  sura  of  all  the  terms. 

CASE    I. 

395.  Having  given  the  first  term,  the  common  difference,  and 
the  number  of  terms,  to  find  the  last  term. 

1.  The  first  term  of  an  increasing  progression  is  4,  the  com- 
mon difference  3,  and  the  number  of  terms  10  :  what  is  the 
last  term? 

Analysis. — Bj  considering  the  manner  in  operation. 

which  the  increasing  progression  is  formed,  9       No.  less  1 

we  see  that  the  2d  term  is  obtained  by  add-         _3       com.  diff. 
ing  the  common  difierence  to  the    1st  term;  2t 

the  3d,  by  adding  the  common  diflference   to  4        1st  term, 

the  2d;   the  4th,  by  adding  the  common  dif-         ^       Yast  term 
ference  to  the   3d,  and    so  on;    the  number 

of  additions^  in  every  case^  heing  one  less  than  the  number  of 
terms  found.  Instead  of  making  the  additions,  we  may  multiply 
the  common  difference  by  the  number  of  additions,  that  is,  by  1 
less  than  the  number  of  terms,  and  add  the  first  term  to  the  pro- 
duct. 

Rule. — Multiply  the  common  difference  hy  1  less  than 
the  number  of  terms :  if  the  progression  is  increasing^  add 
the  product  to  the  first  term,  and  the  sum  will  he  the  last 
term  /  if  it  is  decreasing,  subtract  the  product  fro7n  the 
first  term,  and  the  difference  will  be  the  last  term. 

Examples. 

1.  What  is  the  18th  term  of  an  arithmetical  progression,  d 
which  the  first  term  is  4,  and  the  common  diflference  5  ? 

2.  A  man  is  to  receive  a  certain  sum  of  money  in  12  pay- 
ments :   the  first   payment   is   $800,  and   each   succeeding   pay- 


ARITHMETICAL    PROGRESSION.  841 

ment  is  less  than  the  previous  one  by  $20  :    what  will  be  the 
last  payment  ? 

3.  What  will  $200  amount  to  in  15  years,  at  simple  in- 
terest, the  increase  being  $14  for  the  first  year,  $28  for  the 
second,  and  so  on? 

4.  Mr.  Jones  has  12  children.  He  gives,  by  will,  $1000  to 
he  youngest,  $50  more  to  the  next  older,  and   so   on  to  each 

next  older  $50  :  how  much  did  the  oldest  receive  ? 

5.  A  man  has  a  piece  of  land  35  rods  in  length,  which 
tapers  to  a  point,  and  is  found  to  increase  ^  rod  in  width,  for 
every  rod  in  length  :   what  is  the  width  of  the  wide  end  ? 

6.  James  and  John  have  100  marbles.  It  is  agreed  between 
them  that  John  shall  have  them  all,  if  lie  will  place  them  in 
a  straight  line  half  a  foot  apart,  and  so  that  he  shall  be 
obliged  to  travel  300  feet  to  get  and  bring  back  the  furthest 
marble  ;  and  also,  if  he  will  tell,  without  measuring,  how  far 
he  must  travel  to  bring  back  the  nearest.    How  far? 

CASE     II. 

396.  Knowing  the  two  extremes  of  an  arithmetical  progres- 
sion, and  the  number  of  terms,  to  find  the  common  diflference. 

1.  The  two  extremes  of  a  progression  are  4  and  68,  and 
the  number  of   terms  17  :    what  is  the  common  diflference? 

Analysis, — Since  the  common  difference  operation. 

multiplied  by   1   less  than  the   number  of  68 

terms  gives  a  product  equal  to  the  differ-  4 

ence  of  the  extremes,  if  we  divide  the  dif-     lY  _  i  ==  16)64(4 
ference  of  the  extremes  by  1  less  than  the 

number   of  terms,    the    quotient   will    be    the  common    difference: 
hence, 

Rule. — Subtract  the  less  extreme  from,  the  greater^  and 
divide  tliG  remainder  by  1  less  than  the  7nimber  of  terms: 
the  quotient  will  be  the  common  difference. 


34:2  ARITHMETICAL   PROGRESSION. 


Examples. 

1.  A  man  started  from  Chicago  and  traveled  15  days  ;  each 
day's  journey  was  longer  than  that  of  the  preceding  day  by 
the  distance  which  he  traveled  the  first  day:  what  was  his 
daily  increase  if  he  traveled  15  miles  the  last  day? 

2.  A  merchant  sold  14  yards  of  cloth,  in  pieces  of  1  yard 
each  ;  for  the  first  yard  he  received  $^,  and  for  the  last  $2  6 J  : 
what  was  the  difference  in  the  price  per  yard? 

3.  A  board  is  17  feet  long  ;  it  is  2  J  inches  wide  at  one 
end,  and  14|-  at  the  other  :  what  is  the  average  increase  in 
width  per  foot  in  length? 

4.  The  fourth  term  of  a  series  is  12,  and  the  eleventh  is 
i3  :  find  the  intermediate  terms. 

CASE    III. 

397.  To  find  the  sum  of  the  terms  of  an  arithmetical  progres- 
sion. 

1.  What  is  the  sum  of  the  series  whose  first  term  is  2, 
common  difference  3,  and  the  number  of  terms  8  ? 


Given  series,  )^  ^  ^        ^^        ^^ 

Same,  order  inverted,  [- 23        20        It        14        11 
Sum  of  both  series,     )  25  +  25+25  +  25+25 


Given  series.  'i     2  5  8        11        14        It       20     23 

8      _5     ^ 
25  +25  +  25+25  +  25+25  +  25 


Analysis. — The  two  series  are  the  same;  hence,  their  sum  i3 
equal  to  twice  the  given  series.  But  their  sum  is  equal  to  the 
sum  of  the  two  extremes,  2  and  23,  taken  as  many  times  as  there 
are  terms ;  and  the  given  series  is  equal  to  half  this  sum,  or  to  the 
sum  of  the  extremes  multiplied  by  half  the  number  of  terms. 

Rule. — Add  the  extremes  together^  and  multiply  their  sum 
by  half  the  number  of  terms  y  the  product  will  be  the  sum 
of  aU  the  terms. 


ARITHMETICAL    PROGRESSION.  343 

Examples. 

1.  What  debt  could  be  discharged  in  a  year,  by  weekly 
payments  in  arithmetical  progression,  the  first  payment  being 
15,  and  the  last  $100? 

2.  A  person  agreed  to  build  56  rods  of  fence  ;  for  the  firsfc 
rod  he  was  to  receive  6  cents,  for  the  second,  10  cents,  and 
so  on :  what  did  he  receive  for  the  last  rod,  and  how  much 
for  the  whole? 

3.  If  a  person  travels  30  miles  the  first  day,  and  a  quarter 
of  a  mile  less  each  succeeding  day,  how  far  will  he  travel  in 
30  days? 

4.  If  120  stones  be  laid  in  a  straight  line,  each  at  a  dis- 
tance of  a  yard  and  a  quarter  from  the  one  next  to  it,  how 
far  must  a  person  travel  who  picks  them  up  singly  and  places 
them  in  a  heap,  at  the  distance  of  6  yards  from  the  end  of 
the  line  and  in  its  continuation  ? 

CASE     IV. 

398.  Having  given  the  first  and  last  terms,  and  the  common 
diflFerence,  to  find  the  number  of  terms. 

1.  The  first  term  of  an  arithmetical  progression  is  5,  the 
common  difference  4,  and  the  last  term  41 ;  what  is  the  number 
of  tenns  ? 

Anaj  rsis. — Since  the  last   term    is  opeeation. 

equal  to  the  first  term  added  to  the  41  —  5  =  36 

product  of  the  common  difference,  by  4)36(=9 

one  less  than  the  number  of  terras  0  -*-  1  =  10  x7o.  terms 
(Art.  395),  it  follows  that,  if  the  first 

term  be  taken  from  the  last  term,  the  difference  v/ill  be  equal  to 
the  product  of  the  common  difference  by  1  less  than  the  number 
of  terras :  if  this  be  divided  by  the  common  difference,  the  quotient 
will  be  1  less  than  the  number  of  terms. 


344  GEOMETRICTL   PllOGRESSION. 

Rule. — Divide  the  difference  of  the  two  extreoiies  by  the 
common  diffei^ence,  and  add  1  to  the  quotient:  the  sum  will 
be  the  number  of  terms. 

Examples. 

1.  A  fanner  sold  a  number  of  bushels  of  wheat ;  it  was 
agreed  that,  for  the  first  bushel,  he  should  receiye  50  cents, 
and  an  increase  of  9  cents  for  each  succeeding  bushel,  and  for 
the  last,  he  received  $500  :  how  many  bushels  did  Le  sell  ? 

2.  A  person  proposes  to  make  a  journey,  and  to  travel  15 
miles  the  first  day,  and  33  miles  the  last,  with  a  daily  in- 
crease of  1^  miles  ;  in  how  many  days  did  he  make  the  jour- 
ney, and  what  was  the  whole  distance  traveled? 

3.  I  owe  a  debt  of  12325,  and  wish  to  pay  it  in  equal  in- 
stallments, the  first  payment  to  be  $575,  the  second,  1500, 
and  decreasing  by  a  common  difference,  until  the  last  payment, 
which  is  $200  :  what  will  be  the  number  of  installments? 


G-EOMETRICAL     PROG-RE  SSI  ON. 

399.  A  Geometrical  Progression  is  a  series  of  terms,  each 
of  which  is  derived  from  the  precedmg  one,  by  multiplying  it 
by  a  constant  number.  The  constant  multiplier,  is  called  the 
ratio  of  the  progression. 

400.  An  increasing  series  is  one  whose  ratio  is  greater 
than  1  : 

A  decreasing  series  is  one  whose  ratio  is  less  than  1. 
rhus, 

1,     2,     4,     8,    16,   32,  &c. — ratio  2 — increasing  series  : 
32,  16,    8,     4,     2,     1,    &c. — ratio  ^ — decreasing  series.       i 

The  several  numbers  resulting  from  the  multiplication,  are 
called  terms  of  the  progression.     The  first   and  last  terms  are 


GEOMETRICAL  PROGRESSION.  345 

called    the   extremes,    and   the    intermediate    terms    are    called 
means. 

401.  In  every  Geometrical,  as  well  as  in  every  Arithmetical 
Progression,  there  are  five  parts  : 

1st,   The  first  term  ; 
2d,    The  last  term  ; 
3d,    The  common  ratio  ; 
4  th,  The  number  of  terms  ; 
5th,  The  sum  of  all  the  terms  ; 
If  any  three  of  these  parts  are  known,  or  given,  the  remain- 
ing ones  can  be  determined. 

CASE     I. 

402.  Having  given  the  first  term,  the  ratio,  and  the  number 
of  terms,  to  find  the  last  term. 

1.  The  first  term  is  4,  and  the  common  ratio  3  :  what  is 
the  5th  term? 

Analysis. — The    second    term    is  operation. 

formed  by  multiplying  the  first  term         3x3x3x3=81 

by    the    ratio;    the    third   term,    by  4 

multiplying  the  second  terra  by  the  Ans.   324 

ratio,    and    so    on;     the    number    of 

multiplications  leing  1  less  than  the  number  of  terms:  thus, 

4  =      4,  1st  term, 
3x4=    12,  2d  term, 
3x3x4=    3G,  3d  term. 
3x3x3x4  =  108,  4th  term. 
3x3x3x3x4  =  324,  5th  term. 

Therefore,  the  last  term  is  equal  to  the  first  term  multiplied 
y  the  ratio  raised  to  a  power  whose  exponent  is   1   less  than 
the  number  of  terms. 

Rule. — Baise  the  ratio  to  a  power  whose  exponent  is  1 
less  than  the  number  of  terms,  and  then  multiply  this  power 
by  the  first  term. 


L 


846  GEOMETRICAL   PKOGRESSION. 

Examples. 

1.  The  first  term  of  a  decreasing  progression  is  2187  ;  the 
ratio  is  J,  and  the  number  of  terms  8  :  what  is  the  last 
term  ? 

2.  The  first  term  of  an  increasing  geometrical  series  is  8 
ihe  ratio  5  :   what  is  the  9th  term  ? 

3.  The  first  term  of  a  decreasing  geometrical  series  is  729, 
the  ratio  J  :    what  is  the  10  th  term  ? 

4.  If  a  farmer  should  sell  15  bushels  of  wheat,  at  1  mill 
for  the  first  bushel,  1  cent  for  the  second,  1  dime  for  the 
third,  and  so  on  ;   what  would  he  receive  for  the  last  bushel  ? 

5.  A  man  dying  left  5  sons,  and  bequeathed  his  estate  in 
the  following  manner  :  to  his  executors,  $100  ;  to  his  youngest 
sou  twice  as  much  as  to  the  executors,  and  to  each  son 
double  the  amount  of  the  next  younger  brother :  what  was 
the  eldest  son's  portion  ? 

6.  A  merchant  engaging  in  business,  trebled  his  capital  once 
ill  4  years  :  if  he  commenced  with  $2000,  what  would  his  capital 
amount  to  at  the  end  of   the  12th  year? 

7.  A  farmer  wishing  to  buy  16  oxen  of  a  drover,  finally 
agreed  to  give  him  for  the  whole  the  cost  of  the  last  ox  only. 
He  was  to  pay  1  cent  for  the  first,  2  cents  for  the  second, 
and  doubling  on  each  one  to  the  last :  how  much  would  they 
cost  him  ? 

8.  What  is  the  amount  of  $500  for  3  years  at  6  per  cent, 
compound  interest? 

Note.— The  ratio  is  1.06. 

CASE     II. 

403.  Knowing  the  two  extremes  and  the  ratio,  to  find  the 
sum  of  the  terms. 

1.  What  is  the  sum  of  the  terms  of  the  progression  2,  6, 
18,  54,  162? 


GEOMETRICAL    PROGRESSION.  34:7 

OPLRATIOX. 

6   1-  18  +  54  +  162  +  486  =  3  times. 
2  +  6  +  18  +  54  +  162  =1  time. 

486  -  2  =  2  times 

486-2       484 


2  2 


=  242  sum. 


Analysis.  —  If  we  multiply  the  terms  of  the  progression  hy 
tlie  ratio  3,  we  have  a  second  progression,  6,  18,  54,  162,  486, 
Avhich  is  3  times  as  great  as  the  first.  If  from  this  we  subtract 
the  first,  the  remainder,  486  —  2,  will  be  2  times  as  great  as  the 
first;  and  if  this  remainder  be  divided  by  2,  the  quotient  will  be 
the  sum  of  the  terms  of  the  first  progression. 

But  486  is  the  product  of  the  last  term  of  the  given  progression 
multiplied  by  the  ratio;  2  is  the  first  term;  and  the  divisor  2,  1 
less  than  the  ratio :   hence, 

Rule. — Multiply  the  last  term  by  the  ratio;  take  the  dif- 
ference hetiveen  this  product  and  the  first  term,  and  divide 
the  remainder  by  the  difference  between  1  and  the  ratio. 

Note. — When  the  progression  is  increasing,  the  first  term  is  sub- 
tracted from  the  product  of  the  last  term  by  the  ratio,  and  the  di- 
visor is  found  by  subtracting  1  from  the  ratio.  When  the  progression 
is  decreasing,  the  product  of  the  last  term  by  the  ratio  is  subtracted 
from  the  fii'st  term,  and  the  ratio  is  subtracted  from  1. 

Examples. 

1.  The  first  term  of  a  progression  is  4,  the  ratio  3,  and  the 
last  term  *I8t22  :   what  is  the  sum  of  the  terms? 

2.  The  first  term  of  a  progression  is  1024,  the  ratio  \,  and 
the  last  term  4  :   what  is  the  sum  of  the  series  ? 

3.  What  debt  can  be  discharged  in  one  year  by  monthly 
payments,  the  first  being  ^2,  the  second  $8,  and  so  on  to  the 
end  of  the  year  ;  and  v,hat  will  be  the  last  payment  ? 

4.  A  gentleman   being    impartuued  to  sell  a  fine  horse,  said 


348  ANALYSIS. 

that  he  would  sell  him  on  the  condition  of  receiving  1  cent 
for  the  first  nail  in  his  shoes,  2  cents  for  the  second,  and  so 
on,  doubling  the  price  of  every  nail :  the  number  of  nails  in 
each  shoe  being  8,  how  much  would  he  receive  for  his  horse? 
5.  A  laborer  agreed  to  thresh  64  days  for  a  farmer,  on  the 
condition  that  he  should  give  him  1  grain  of  wheat  for  the 
first  day's  labor,  2  grains  for  the  second,  and  double  each 
succeeding  day :  what  number  of  bushels  would  he  receive, 
supposing  a  pint  to  contain  7680  grains ;  and  what  number  oi 
ships,  each  carrying  1000  tons  burden,  might  be  loaded,  allow- 
ing 40  bushels  to  a  ton? 


ANALYSIS. 

404.  An  Analysis  is  an  examination  of  the  separate  parts 
of  a  proposition,  and  of  the  connection  of  those  parts  with 
each  other. 

In  analyzing,  we  generally  reason  from  a  give?!  number  to 
its  unit,  and  then  from  this  unit  to  the  required  number. 

The  process  is  indicated  by  the  relations  which  exist  between 
the  given  and  the  required  numbers,  and  pursued,  step  by  step, 
independently  of  set  rules. 

1.  If  12  yards  of  cloth  cost  $48.36,  what  will  1  yards  cost  ? 

Analysis. — One  yard  of  cloth  will  cost  j\  as  much  as  12  yards : 
since  12  yards  cost  $48.36,  one  yard  will  cost  ^\  of  $48.36  =  $4.03 ; 
7  yards  will  cost  7  times  as  much  as  1  yard,  or  7  times  y\  of 
$48.36  =  $28.21 ;  therefore,  if  12  yards  of  cloth  cost  $48.36, 
7  yards  will  cost  $28,21. 

OPEEATION. 

Jj  of  48.36  =  $4.03  =  price  of  1yd.,        j  48.36  x  '?_.^gp  g, 
4.03   X    1  =  $28.21  =  price  of  1yd.,  ^''  (       12         -^^^'^^ 

2.  If  21  pounds  of  butter  will  buy  45  pounds  of  sugar,  how 
much  butter  will  36  pounds  of  sugar  buy  ? 


ANALYSIS.  349 

Analysis.— One  pound  of  sugar  will  buy  ^^  of  271b.  of  butter, 
and  861b.  of  sugar  will  buy  36  times  ,V  of  271b. 

OPERATION. 

^^j  of  27  =  fl  =  value  of  1  lb.  of  sugar,      j  27  X  36  _  oi  3  ik 
-27  X  36  =21f  lb.  =  value  of  361b.  "      ^"^  \       45       ~      *     * 

3.  What  will  6f  cords  of  wood  cost,  if  2f  cords  cost  %^t 

Analysis. — Price  divided  by  quantity,  or  7|  -r-  2^  =  y  -r  V  == 
$3  =  price  of  1  cord;   $3  x  6|  =  $201  =  cost  of  6f  cords. 

OrERATION. 

7i-^2|x6J  =  -x  —  X—  =  ^  =  201-  =  $20.25.  Ans. 
V  *        8        19        4         4  ^ 

4.  A  farmer  sold  a  number  of  cows,  and  had  12  left,  which 
was  J  of  the  number  sold  ;  if  the  number  sold  be  divided  by 
I  of  9J,  the  quotient  will  be  \  the  number  of  dollars  he 
received  per  head  :  how  much  did  he  receive  per  head  for  his 
cows  ? 

Analysis.— 12  is  ^  of  3  times  12  =  36;  36 -r- 1  of  9J  =  36 
y.  \  =  o\  =i\  of  the  price  per  head;  5|  x  5  =■  $25 f  =  price  per 
head. 

OPERATION. 

12X3 --3  of  9Jx5=i^^  xi  of  A  x  5 :,,  i|o  ^ |25f 

5.  What  will  20  bushels  of  barley  cost,  in  dollars  and  cents, 
at  7  shillings  a  bushel,  New  York  currency  ? 

Analysis.  —  Twenty    bushels    will    cost    7x20  =  140s.      and 
40  -r  8  =  $17.50,  since  in  New  York  8s.  =  1  dollar. 

OPERATION. 

!iL?2  =  $n.5o.  Am. 
8 


350  ANALYSIS. 

6.  Wheat  will  12 J  pounds  of  tea  cost  at  6s.  8d.  a  pound, 
Pennsylvania  currency  ? 

Analysis.  —  6s.  8d.  =  80d. ;  80  x  12^  =  lOOOd.  =  price  in 
pence  of  12|lbs, ;  and  since  7s.  6d.  =  90d.  Pennsylvania  currency, 
equal  1  dolfar,  1000  —  90  =  $11.11^  =  cost  of  121  lb.  in  Federal 
money. 

OrERATION. 

80j<J2i^8^X^^ 

90  90  X  2  ^  ^ 

1.  How  many  days'  work,  at  10s.  6d.  a  day,  must  be  given 
for  18  bushels  of  corn  at  5s.  lOd.  a  bushel? 

Analysis. — 18  bushels  are  worth  5s.  lOd.  x  18  =  70d.  x  18  = 
12G0d. ;  and  for  this  as  many  days'  labor  must  be  given  as 
10s.  6d.  =  126d.  is  contained  times  in  1260d.:  1260  -r  126  =  10 
days. 

OPERATION. 

5s.  lOd.  =    TOd.  ;       70  x  is 
10s.     6d.  =  126d.;       -T^T- =  ^^  ^^y''    ^^^- 


8.  A  merchant  bought  a  number  of  bales  of  cloth,  each 
containing  133 J  yards,  at  the  rate  of  12  yards  for  $11,  and 
sold  it  at  the  rate  of  8  yards  for  $t,  by  which  he  lost  $100 
in  the  trade  :  how  many  bales  were  there  ? 

Analysis.— One  yard  costs  j^  of  $11  =  $11;  one  yard  was 
Bold  for  I  of  $7  =  $^.  The  difference  between  ii  and  |,  which  is 
$2*4,  is  the  loss  on  each  yard. 

Since  the  total  loss  was  $100,  he  must  have  sold  as  many  yards 
s  $Jj-  is  contained  times  in  $100. 

OPERATION. 

H  —  I  =  $2T  =  loss  per  yard. 
100  -^  Jj  =  2400  yds. ;         2400  -~  1331  =  18  bales.    Ans. 

Proof. 

100    .     1     .    ,001        100       24  3  ,Q  ,    ,  . 

— -  -. lodi  =  —  X  —  X =  18  bales.    Ans, 

1         24  11        400 


ANALYSIS.  351 

9.  A  can  mow  an  acre  of  grass  in  7  ^  hours  ;  B,  in  5  hours : 
0,  in  5f  hours :  how  many  days,  working  6  J  hours,  would 
they  require  to  mow  13|-  acres? 

Analysis. — Since  A  can  mow  an  acre  of  grass  in  7^  hours,  B 
in  5  hours,  and  0  in  5|,  A  can  niow  y^j,  B,  |,  and  C,  ^%  of  an 
acre  in  1  hour.  Together,  tliey  can  mow  y^^  ^  i  _^  ^e^  —  23  ^f  ^^ 
acre  in  one  hour;  and  to  mow  13i  acres,  they  will  require  as 
many  hours  as  f§  is  contained  times  in  13|,  which  is  27;  and 
working  6J  hours  each  day,  it  will  take  as  many  days  as  6^  is 
contained  in  27;    27-4-0^  =  4  days;   hence,  it  will  require  4  days. 

OPKRATION. 

^\  +  J  -\-  ^5  =  Jl  acres  =  what  they  all  do  in  1  hour. 

G9       45        4 
13t  -^  -2|  -^  G|  =  —  X  —  X  —  =  4  days.    Ans. 
*       ^^         *        5        23       27 

10.  A  person  employed  three  men,  A,  B,  and  C,  to  do  a 
piece  of  work  for  $132.G6.  A  can  do  the  work  alone  in  23 J 
days,  working  12  hours  a  day  ;  B  can  do  it  in  25  days,  work- 
ing 8  hours  a  day;  and  C  can  do  it  in  16  days,  working  Hi 
liours  a  day.  In  what  time  can  the  three  do  it,  working  to- 
gether, 10  hours  a  day,  and  what  share  of  the  money  should 
each  receive  ? 

Analysis. — Since  A  can  do  the  work  in  23}  days,  working  12 
hours  each  day;  B,  in  25  days,  working  8  hours  each  day;  and 
C,  in  16  days,  working  11|  hours  each  day;  A  can  do  the  work 
in  280  days;  B,  in  200  days;  and  0,  in  180  days;  working  1  hour 
each  day.  Then  A,  B,  and  0,  can  do  2I5  +  a^ir  +  riff  =  tb^ih 
of  the  work  in  1  day,  working  1  hour.  By  working  10 
hours  they  will  do  10  times  as  much;  or,  the  work  done  hy  each 
in  1  day  of  10  hours,  will  he  denoted  by  oVtfj  u'Aj  ^^^  tA?  and 
the  whole  work  done  in  1  day  by  gWiii  hence,  the  number  of 
days  will  be   denoted  by  the   number  of  times  which  1   contains 

2020—    ^56    —  'TI9"'^3S- 

If  the  part  which  each  does  in  1  day  be  multiplied  by  the 
number  of  days,  viz.,  7^''g,  the  product  will  be  the  part  done  by 


352  ANALYSIS. 


h;    viz.,     A,     -i^o  X  7A  =  iV.;     B,  ^Vo  X  T/9  =  A\; 

and 

tVo  X  'J's'V  =  TT5 ;   therefore,   A  must    have    jS%;    B,  y\\; 

and 

j-W  of  $132.66. 

OPERATION. 

2V0  +  w  +  T8^o  =  tMo  =  tte  work  done  in  one  day. 

1  -^  yWo  =  *^¥9  ^^ys,  in  which  the  three  do  the  work. 

^H%  X  "^¥9  =  TT8»  and  yVs  0^  132.66  =  $33.53ff,    A's  share 

2V0  X  7/9  =  TTs  :         t¥b  of  132.66  =  $46.95||,    B's  share 

i¥o  X  73V  =  tVs  :         tW  of  132.66  =  $52.16|f,    C's  share. 

11.  If  336  men,  in  5  days,  working  10  hours  each  day. 
can  dig  a  trench  of  5  degrees  of  hardness,  70  yards  long,  3 
yards  wide,  and  2  yards  deep  ;  how  many  days,  of  12  hours 
each,  will  240  men  require  to  dig  a  trench  36  yards  long,  5 
yards  wide,  and  3  yards  deep,  of  6  degrees  of  hardness  ? 

Analysis. — The  first  trench  may  be  represented  by  the  product 
of  its  elements,  5  x  70  x  3  x  2 ;  and  the  second  by  the  product 
of  its  elements,  6x36x5x3.  Inl  day  of  1  hour,  one  man 
can  do  5^^  of  |  of  y^  of  5  x  70  x  3  x  2 ;  and  240  men,  in  one 
day  of  12  hours,  can  do  240  x  12  times  as  much  ; 
or,    210  ^   i_2  ^  ^1^  of  1    Qf  j^  Q^   5X70X3X2  =  360  =  work 

done  in  one  day  by  240  men.  It  will  take  as  many  days  to  dig 
the  second  trench  as  360  is  contained  times  in  6x36x5x3. 
6  X  36  X  5  X  3-f- 360  =  9  days. 

OPERATION. 

6x36x5x3-^^f^XVX3i^of  Jof  jLof  5_XJ^_x_3_x^^9. 

12.  If  20  cords  of  wood  are  equal  in  value  to  6  tons  of 
hay,  and  5  tons  of  hay  to  36  bushels  of  wheat,  and  12  bush- 
els of  wheat  to  25  bushels  of  com,  and  14  bushels  of  com  to 

6  pounds  of  butter,  and  72  pounds  of  butter  to  8  days  Ox 
labor  ;  how  many  cords  of  wood  will  be  equal  to  16  days  of 
labor  ? 

Analysis.— 1  day's  labor  is  worth  |  of  72  lb.  of  butter  =  9  lb. 
1  lb.  of  butter  =  ^V  of  14  bush,  corn  =  ^^  =  1  bush,  corn,  and  9  lb. 


ANALYSIS.  353 

are  worth  |  x  9  =  |  bush.  corn.  1  bush,  corn  is  worth  -^j  of  12 
bush,  wheat  =  ^f  bush.,  and  |  bush,  corn  are  worth  |f  x  4=1^ 
bush,  wheat  1  bush,  wheat  is  worth  ^\  of  5  tons  hay  =  ^^^  tons, 
and  I J  bush,  are  worth  ^%  x  11  =  2^^  tons  of  hay.  1  ton  is  worth 
I  of  20  cords  =  %^  cords,  and  o^  tons  =  ^  x  /j  =  ^  cord.  This 
is  the  value  of  one  day's  abor,  and  16  days'  labor  will  be  worth 
16  times  I  cord  =  8  cords. 

OPERATION. 

i  of  V  of  li  of  if  of  /g  of  2^0  X  V^  =  8  cords.     Ans. 
(See  Art.  3t0.) 

13.  A,  B,  and  C,  put  in  trade  $5626  :  A's  stock  was  in 
5  months,  B's,  7  months,  and  C's,  9  months.  They  gained 
$1260,  which  was  so  divided  that  A  received  $4  as  often  as 
B  had  $5,  and  as  often  as  C  had  $3.  After  receiving 
$2164.50,  B  absconded  :  what  was  each  one's  stock  in  trade, 
and  how  much  did  A  and  C  gain  or  lose  by  B's  withdrawal? 

Analysis. — Since  A  received  $4  as  often  as  B  had  $5,  and  as 
often  as  C  had  $3,  if  tlie  whole  gain  were  divided  into  12  equal 
parts,  A  would  have  y*2,  B,  y\,  and  0,  fW,  of  $1260;  or  A  would 
have  $420,  B,  $525,  and  C,  $315.  Now,  if  their  respective  gains 
bo  divided  by  the  number  of  months  each  one's  stock  continued 
in  trade,  the  quotients  will  represent  their  monthly  gains,  viz.,  A's 
will  be  420  -^  5  =  $84;  B's,  $525  -^  7  =  $75 ;  and  C's,  $315  ~  9  = 
$35,  which  gives  $194  as  their  whole  gain  for  1  month. 

But  since  each  one's  share  of  the  gain  for  a  given  time  will  be 
to  the  whole  gain  for  the  same  time,  as  each  one's  stock  to  the 
whole  stock;  it  follows,  that  A  will  have  y^^,  B,  yY,^,  and  0,  y'^^^, 
of  the  whole  stock,  or  A  will  have  $2436,  B,  $2175,  and  0,  $1015. 
When  B  ran  away  he  was  entitled  to  his  original  stock,  $2175,  and 
his  share  of  the  gain  for  7  months,  that  is,  to  $2175  -\-  $525  = 
$2700;  but  as  he  took  away  only  $2164.50,  A  and  0  gained 
$535.50  by  his  withdrawal,  which  must  be  divided  between  them 
in  the  ratio  of  their  investments,  or  as  4  to  3;  therefore,  A  will 
have  ^,  and  0  ^  of  B's  unclaimed  portion,  or  A  will  have  $306, 
and  C  $229.50. 


354  ANALYSIS. 

OPERATION. 

4  +  5  +  3  =  12. 

A's  whole  gain  =  ^^  of  $1260  =  $420 

B's       "         "    =-^^  "      "       =1525 

C's       "         "    =  ^2  "      *'       =1315 

A's   monthly  gain  =  $420  -^  5  =  $84 
B's        "  "    =  $525  -^  T  =  $75 

C's         "  "    =  $315  -T-  9  =  $35 

$194 

A's  stock  =  /y^  of  $5626  =  $2436 
B's  "  =  j-7_5_  u  u  ^  ^2175 
C's      "      =  /^   "       "      =  $1015 

$2175  +  $525  -  $2164.50  =  $535.50,  what  B  left. 
4-  of  $535.50  =  $306,        A's  share  of  it. 
3    "         "       =  $229.50,  B's  share  of  it. 

14.  Mr.  Johnson  bought  goods  to  the  amount  of  $2400,  \ 
to  be  paid  in  3  months,  |-  in  4  months,  -J-  in  6  months,  and 
the  remainder  in  8  months  :  what  is  the  equated  time  for  the 
payment  of  the  whole  ? 

Analysis. — $800  to  be  paid  in  3  months,  is  the  same  as  $1  to 
be  paid  in  2400  months;  $C00,  in  4  months,  the  same  as  $1  in 
2400  months;  $G00,  in  6  months,  the  same  as  $1  in  8600  months; 
and  $400  in  8  njonths,  the  same  as  $1  in  3200  months.  Then 
$1,  payable  in  2400  +  2400  +  3600  +  3200  =  11600  months,  is  the 
same  as  $2400  in  ^^Vo  of  H^OO  months,  which  is  4|  months  =  4 
months  25  days,  the  equated  time  of  payment. 

OrERATION. 

800  X  3  =  2400 
600  X  4  =  2400 
600  X  6  =  3600 
400  X  8  =  3200 


2400      11600 
11600  ~  2400  =  4f  mo.  =  4  mo.  25  da.  Ans. 


ANALYSIS.  355 

15.  What  will  be  the  interest  on  $60.48  for  1  year  3 
months,  at  7  per  cent.? 

Analysis. — Since  the  interest  on  $1  for  1  year  is  7  cents,  or 
seven  hundredths  of  $1,  the  interest  on  $00.48  for  1  year,  will 
be  $60.48  x  .07  =  $4.2336.  The  interest  for  1  month  will  be  j\ 
as  much  as  for  1  year,  or  j\  of  $4.2336  =  $0.3528;  and  for  lyr 
3  mo.  =  15  months,  it  will  be  15  times  as  much  as  for  1  month, 
or  $0.3528  x  15  =  $5,292. 

OPEBATION. 

($60.48  X  .07  -^  12)  X  15  =  $5,292.     Ans. 

16.  What  will  be  the  interest  on  $88.92,  for  8  mo.  20  da., 
at  7  per  cent.? 

17.  A  merchant  has  three  kinds  of  cloth,  worth  $lf,  $2^, 
'^^f,  a  yard  :  what  is  the  least  number  of  whole  yards  he  can 
sell,  to  receive  an  average  price  of  $2^  a  yard? 

Analysis. — If  he  sells  1  yard  worth  $lf,  for  $2^,  he  will  gain 
f  of  a  dollar;  to  gain  1  dollar,  he  must  sell  as  many  yards  as  |  is 
contained  times  in  1,  or  ^  yards.  But  since  he  is  neither  to  gain 
nor  lose  by  the  operation,  if  he  gains  on  one  kind,  he  must  lose  an 
equal  sum  on  some  other;  hence,  he  must  sell  some  that  is  worth 
more  than  the  average  price.  If  he  sells  1  yard  worth  $3 J  for  $2i, 
lie  will  lose  |  of  a  dollar;  and  to  lose  $1,  he  must  sell  |  of  a  yard. 
Therefore,  to  make  the  loss  equal  to  the  gain,  he  must  sell  i  of  a 
yard  at  $3^  a  yard,  as  often  as  he  sells  f  of  a  yard  at  $1^  a 
yard. 

If  he  sells  1  yard  worth  $2},  for  $2,},  he  gains  |  of  a  dollar, 
and  to  gain  $1  he  must  sell  4  yards;  hence,  to  keep  the  average 
price,  he  must  lose  as  much  on  some  other;  and  as  he  can  only 
lose  on  that  at  $3J  a  yard,  he  must  sell  enough  of  that  to  lose 
$1,  which  would  be  ^  of  a  yard;  therefore,  as  often  as  he  sell 
I  yard  at  $1§  a  yard,  he  must  sell  |  yard  at  $3 J  a  yard;  and 
as  often  as  he  sells  4  yards  at  $2|  a  yard,  he  must  sell  |  yard 
at  $3  J  a  yard. 

But  since  it  is  desirable  to  have  the  proportional  parts  expressed 
in  the  least  whole  numbers,  we  may  multiply  the  numbers  by  the 


356 


ANALYSIS. 


least  common  multiple  of  their  denominators,  and  divide  the  pro- 
ducts by  their  greatest  common  factor;  this  being  done,  we  obtain 
in  the  above  example,  3  yards  at  $lf  a  yard,  10  yards  at  %2l  a 
yai-d,  and  4  yards  at  $3|  a  yard. 


OPERATION. 


H 


If  ] 

1 

6 

6 

3fJJ 

4 

20 

20 

* 

1- 

4 

4 

8 

3 

10 

4 


18.  The  hour  and  minute  hands  of  a  clock  are  together  at 
12  o'clock:   when  are  they  next  together? 

Analysis. — Since  the  minute-hand  passes  over  60  minute  spaces 
while  the  hour-hand  passes  over  5,  the  minute-hand  passes  over  12 
minute  spaces  while  the  hour-hand  passes  over  1,  gaining  11 
minute  spaces  on  the  hour-hand  in  12  minutes  of  time,  the 
minute-hand  requiring  one  minute  of  time  to  pass  over  1  minute 
of  space.  Hence,  in  1  minute  of  time,  the  minute-hand  gains  on 
the  hour-hand  ^  of  a  minute  space. 

When  the  minute-hand  has  returned  to  12,  the  hour-hand  will 
be  at  1,  and  the  minute-hand  has  then  to  gain  5  minute  spaces.  As 
the  minute-hand  gains  \^  spaces  in  1  minute  of  time,  it  will  take 
as  many  minutes  as  ||  is  contained  times  in  5,  viz.,  5y\  mi.  = 
5  mi.  27y^  sec,  which  added  to  1  o'clock,  gives  1  hr.  5  mi.  27-/V  sec. 

Second  Analysis.  —  In  12  hours  the  minute-hand  passes  the 
hour-hand  11  times;  consequently,  if  both  are  at  12,  the  minute- 
hand  will  pass  the  hour-hand  the  first  time  in  ^  of  12  hours,  or 
Ihr.  5  mi.  27fVsec.  It  will  pass  it  the  second  time  in  ^j  of  12 
hours,  and  so  on. 

OPERATION. 

5  X  XT  =  IT  =  S/jmi.  =  5  mi.  2t^\  sec,  which  added  to 
1  hr.  =  1  hr.  5  mi.  27 jy  sec.     Ans. 

19.  An  apple  boy  bought  a  certain  number  of  apples  at 
the  rate  of  3  for  1  cent,  and  as  many  more  at  4  for  1  cent ; 
and  selling  them  again  at  2  for  1  cent,  he  found  that  he  had 
gained  15  cents  :   how  many  apples  had  he  ? 


ANALYSIS.  357 

Analysis. — Since  ho  bought  a  number  of  apples  at  3  for  a  cent, 
and  as  many  more  at  4  for  a  cent,  Le  paid  ^^  of  a  cent  apiece  for 
the  first,  and  |  of  a  cent  apiece  for  the  second  lot :  then,  J  +  i  =  t2 
of  a  cent,  what  he  paid  for  one  of  each,  and  y'o  -r  2  =  g'V  of  a  cent, 
the  average  price  for  all  he  bought.  Since  he  sold  at  2  for  a  cent, 
or  ^  a  cent  apiece,  he  must  have  gained  on  each  apple  the  diflfer- 
ence  between  ^  and  ^\  =  -^^  of  a  cent ;  hence,  to  gain  1  cent  ho 
must  sell  as  many  apples  as  3?  is  contained  times  in  1  =  4|  apples, 
and  to  gain  15  cents  he  must  sell  15  times  as  many,  or  4f  x  15  =  72 
apples. 

OPBBATION. 

1  -J-  /^f  =  4J,         4f  X  15  =  12  apples.     Ans. 

20.  A  gentleman  left  to  his  three  sons,  whose  ages  were  13, 
15,  and  It  years,  $15000,  to  be  divided  in  such  a  manner, 
that  each  share  being  put  at  interest,  at  1  per  cent.,  should 
give  to  each  son  the  same  amount  when  he  attained  the  age 
of  21  years  :  what  was  the  share  of  each? 

Analysis. — By  the  question,  their  respective  shares  would  be  at 
interest  8,  6,  and  4  years. 

Find   the   present  worth  of  $1   for   8,  6,  and  4   years,  respect- 
ively:    they    are     $0.6410256  +,     $0.7042253  +,     and    $0.78125. 
These  sums  being  put  at  interest  at  7  per  cent.,  will  each  amount 
to  $1  at  the  expiration  of  their  respective  times;   and  the  sum  of 
these  numbers,  $0.6410256  +  $0.7042253  +  $0.78125  =  $2.1265009  + 
is  the  amount,  which  being  so  distributed  among  them,  will  pro- 
duce   $1    to    each.      If   each    number    bo    divided    by    the    sum, 
$2.1265009,  the   quotients  will  denote  the  parts  of  $1,  which,  ac 
cording  to  the  conditions  of  the  question,  each  person  should  re 
ceive;    therefore,   each  person   will   receive   for   his   entire   share 
15000  like  parts  of  one  dollar. 


SoS  ANALYSIS. 


OPERATION. 


81  -r-  1.56  =  $0.6410256  +  present  worth  of  U  for  8  years 

II  -^  1.42  =  $0.^042253  +         ''           "  "  6      " 

$1  -^  1.28  =  10.78125                  "           "  "  4      " 
$2.1265009 

$0.6410256  ~-  2.1265  X  15000  =  $4521.694 


$0.7042253 
$0.78125 


2.1265  X  15000  =  $4967.494 
2.1265  X  15000  =  $5510.815 


21.   A,  B,  C,  and  D,  agree  to  do  a  piece  of  work  for  $312. 

A,  B,  and  C,  can  do  it  in  10  days  ;  B,  C,  and  D,  in  7 J 
days  ;  C,  D,  and  A,  in  8  days  ;  and  D,  A,  and  B,  in  8-|- 
days :  in  how  many  days  can  all  do  it,  working  together ;  in 
how  many  days  can  each  do  it,  working  alone  ;  and  what  part 
of  the  pay  ought  each  to  receive  ? 

Analysis. — Since  A,  B,  0,  can  do  the  work  in  10  days,  they 
can  do  yV  =  iVir  of  it  in  1  day ;  since  B,  C,  D,  can  do  it  in  7^ 
days,  they  can  do  y^^  =  yVV  of  it  in  1  day ;  since  0,  D,  A,  can  do 
it  in  8  days,  they  can  do  |  =  y aV  of  it  in  1  day;   and  since  D,  A, 

B,  can  do  it  in  8 4  days,  they  can  do  ^^  =  j^^  of  it  in  1  day; 
hence,  A,  B,  C,  and  D,  by  working  3  days  each,  will  do 
tVtt  +  t¥o  +  iVv  +  tVo  =  r¥o  of  the  work,  and  in  1  day  they  will 
do  ^  of  y^T^  =  y^a^o  •  I*  '"'ill  then  take  them  as  many  days  to  do  the 
whole  as  y*^^  is  contained  times  in  1  =  6|%  days. 

By  subtracting,  in  succession,  what  the  three  can  do  in  1 
day,  when  they  work  together,  from  what  the  four  can  do  in  1 
day,  we  shall  have  what  each  one  will  do  in  1  day:  viz., 
r\%-T\%=Th,  ^liat  D  will  do  in  1  day;  y^^  _  y^^  =  yf^, 
what  A  can  do  in  1  day;  y^^^  —  j\%  =  tI o»  ^^^^  B  can  do  in  1 
<^ay ;  yVff  -  -iV\  =  yf  05  what  0  can  do  in  1  day.  It  will  take 
each  as  many  days  to  do  the  whole  work  as  the  part  which  h 
can  do  in  one  day  is  contained  times  in  1 :  viz.,  1  ~  yf ^  =  40  days 
A's  time  to  do  it ;  1  -^  y|^  =  30  days,  B's ;  1  -^  yf^  =  24  days 
C's ;     14-  yf  0  =  17|  days,  D's. 

Now,  each  should  receive  such  a  part  of  the  whole  amount  paid, 
viz.,    $312,   as  he   did    of   the    whole    work.     This    part    -^^111    be 


ANALYSIS. 

359 

dvjnoted    by  what   ho 

did  in  1  day  multiplied  bj 

'  tho  number  of 

days   he   wor 

ked:    viz 

•,     A,    rfirX6W  =  ,\; 

B, 

T^  X 

6A  =  t\; 

0,   T^. 

xC/i 

=  /g;     D,  Tizrx6A=TV 

OPERATION. 

A 

~  Tlfe» 

what  A,  B,  C,  do  in 

1 

day. 

ft 

=  t'A. 

"     B,  C,  D,     " 

(( 

1 

=  1*^^. 

"      C,  D,  A,     " 

« 

»'« 

=  iVo, 

"     B,  A,  B,     " 

tl 

iVo 

+  l¥s   +  t¥5 

+  ^\  =  i'^.     what 

A, 

B,   C, 

and  D, 

cau  do  in  3 

days. 

l¥i 

^   3  : 

=  t'A. 

what  A,  B,  C,  and  D, 

,  can  do  in 

1  day. 

i¥o- 

,'!>%  = 

=  xf  (J,    what  A  can  do  m  1  day ; 

1- 

■^72  0' 

=  40da. 

A'o- 

tVo  = 

"  T2^» 

H          J?               11                       It 

1 

^ih 

=  30  da. 

T%- 

AV  = 

"  if  (J» 

li       Q           tl                 tl 

1- 

^ih 

=  24  da. 

T%- 

-  ITSf 

(<        T\            tl                   It 

1 

-Xio 

=  n^da. 

Hence,  the  share  of  each  will  be  : 

$312 

X  A  = 

U9.2Qj%,         A's  share. 

*312 

X  tV  = 

$65.68iV,        B's  share. 

$312 

X  A  = 

$82.10fo,         C^s  share. 

$312 

X  tV  = 

|114.94}J,        D's  share. 

$312.00,  amount  paid  to  A,  B,  C,  and  D 

22.  A  person  owning  §  of  a  vessel,  sold  f  of  his  share  for 
$1736  :  what  was  the  value  of  the  whole  vessel? 

23.  If  a  man  performs  a  journey  in  tj-  days,  traveling  14| 
hours  a  day,  in  how  many  days  will  he  perform  the  same 
ourney  by  traveling  10-J  hours  a  day? 

24.  If  1^  of  a  pole  stands  in  the  mud,  2  feet  in  the  water, 
and  f  above  the  water,  what  is  the  length  of  the  pole  ? 

25.  After  spending  -J-  of  my  money,  and  J  of  the  remainder, 
I  had  $1062  left :  how  much  had  I  at  first  ? 


360  ANALYSIS  AND 

26.  Suppose  a  cistern  has  two  pipes,  and  that  one  can  fill 
it  in  T-i  hours,  and  the  other  in  41-  hours  :  in  what  time  can 
both  fill  it,  running  together? 

21,  If  54  yards  of  ribbon  cost  $9,  what  will  26  yards 
cost? 

28.  If  2  acres  of  land  cost  J  of  f  of  |  of  $300,  what  wil' 
I  of  -J  of  10 J  acres  cost  ? 

29.  A  regiment  of  soldiers,  consisting  of  1000  men,  is  to  be 
clothed  ;  each  suit  is  to  contain  3^  yards  of  cloth  If  yards 
wide  :  how  much  shalloon  that  is  ^  yards  wide  is  necessary  for 
lining  ? 

30.  How  much  tea,  at  Ts.  6d.  a  pound,  must  be  given  for 
234  bushels  of  oats,  at  3s.  9d.  a  bushel,  New  York  currency? 

31.  What  will  3  pipes  of  wine  cost,  at  2s.  9d.  per  quart, 
New  England  currency  ? 

32.  A  gives  B  165  yards  of  cotton  cloth,  at  2s.  6d.  per 
yard,  Missouri  currency,  for  625  pounds  of  lump  sugar  :  how 
much  was  the  sugar  worth  a  pound? 

33.  If  the  expense  of  keeping  1  horse  1  day  is  3s.  4d., 
Canada  currency,  what  will  be  the  expense  of  keeping  4  horses 
3  weeks,  at  the  same  rate? 

34.  Bought  10  bales  of  cloth,  each  bale  containing  14 
pieces,  and  each  piece  22^  yaids,  at  10s.  8d.  per  yard,"-  lUinois 
currency  :  what  was  the  cost  of  the  cloth  ? 

35.  A  has  TJcwt.  of  sugar,  worth  12  cents  a  pound,  for 
which  B  gave  him  12Jcwt.  of  flour:  what  was  the  flour  worth 
a  pound? 

36.  Bought  120  yards  of  cloth,  at  6s.  8d.  a  yard.  New 
York  currency,  and  gave  in  payment  16  bushels  of  rye,  at 
4s.  6d.  a  bushel,  New  England  currency,  and  the  balance  in 
money  :   how  many  dollars  will  pay  the  balance  ? 

31.  A  merchant  bought  21  pieces  of  cloth,  each  piece  con- 
taining 41  yards,  for  which  he  paid  $1260  ;  he  sold  the  cloth 
at  $1.15  per  yard  :   did  he  gain  or  lose,  and  how  much  ? 


PKOMISCL'UUS   EXAMPLES.  361 

38.  The  hour  and  mmute  hands  of  a  watch  are  together  at 
12:   at  what  moment  will  they  be  together  between  5  and  G? 

39.  How  many  yards  of  carpeting  J  of  a  yard  wide  will 
cover  the  floor  of  a  room  18  feet  long  and  15  feet  wide? 

40.  If  9  men  can  build  a  house  in  5  months,  by  working 
12   hours  a  day,  how  many  hours   a  day  must  the  same  mei 

ork  to  do  it  in  6  months? 

41.  B  and  C  can  do  a  piece  of  work  in  12  days  ;  with  the 
assistance  of  A  they  can  do  it  in  9  days  :  in  what  time  can 
A  do  it  alone  ? 

42.  A  can  mow  a  certain  field  of  grass  in  3  days,  B  can 
do  it  in  4  days,  and  C  can  do  it  in  5  days  :  in  what  time 
can  they  do  it,  workmg  together? 

43.  Divide  the  number  480  into  4  such  parts  that  they 
shall  be  to  each  other  as  the  numbers  3,  5,  T,  and  9  ? 

44.  What  length  of  a  board  that  is  8j-  inches  broad,  will 
make  a  square  foot? 

45.  The  provisions  in  a  garrison  were  sufficient  for  1800 
men,  for  12  months  ;  but  at  the  end  of  3  months,  it  was  re- 
inforced by  600  men,  and  4  months  afterward,  a  second  rein- 
forcement of  400  men  was  sent  in :  how  long  would  the 
provisions  last  after  the  last  reinforcement  arrived? 

46.  A  merchant  bought  a  quantity  of  broadcloth  and  baize 
for  $488.80  ;  there  was  11 7  J  yards  of  broadcloth,  at  13^  per 
yard ;  for  every  5  yards  of  broadcloth  he  had  1 J  yards  of 
baize  :  how  many  yards  of  baize  did  he  buy,  and  what  did  it 
cost  him  per  yard? 

4t.  If  the  freight  of  40  tierces  of  sugar,  each  weighmg  3 J 
cwt.,  for  150  miles,  costs  |42,  what  must  be  paid  for  the 
freight  of  10  hhd.,  each  weighing  12 cwt.,  for  50  miles? 

48.  If  1  pound  of  tea  be  equal  in  value  to  50  oranges,  and 
70  oranges  be  worth  84  lemons,  what  is  the  value  of  a  pound 
9f  tea,  when  a  lemon  is  worth  2  cents? 

16 


362  ANALYSIS   AND 

49.  What  amount  must  be  discounted,  at  t  per  cent.,  to 
make  a  present  payment  of  a  note  of  $500,  due  2  years  8 
months  hence  ? 

50.  If  the  interest  on  $225  for  4-J-  years  is  $91.12^,  whafc 
would  be  the  interest  on  |640,  at  the  same  rate,  for  2}  years? 

51.  A  farmer  having  1000  bushels  of  wheat  to  sell,  can 
have  11.15  a  bushel  cash,  or  $1.80  in  nmety  days  :  which 
would  be  most  advantageous  to  him,  money  being  worth  1  per 
cent.  ? 

52.  A  merchant  bought  goods  to  the  amount  of  $1515  on 
9  months'  credit  ;  he  sells  the  same  for  $1800  in  cash  :  money 
being  worth  6  per  cent.,  what  did  he  gain? 

53.  Three  persons  in  partnership  gain  $482.G2  ;  A  put  in  f 
as  much  capital  as  B,  and  B  put  in  f  as  much  as  C  :  what 
was  each  one's  share  of  the  gain? 

54.  A  father  divided  his  estate,  worth  $9268.60,  among  his 
4  children,  giving  A  -J-  of  it,  B  i  and  C  $5  as  often  as  he 
gave  D  $6  :   how  much  did  each  receive  ? 

55.  A  tax  of  $415.50  was  laid  upon  4  villages.  A,  B,  C, 
and  D  ;  it  was  so  distributed,  that  as  often  as  A  and  B  each 
paid  $5,  C  paid  $1,  and  D  $8  ;  what  part  of  the  whole  tax 
did  each  village  pay  ? 

56.  There  are  1000  men  besieged  in  a  town,  with  provisions 
for  5  weeks,  allowing  each  man  16  ounces  a  day.  If  they  are 
reinforced  by  400  men,  and  no  relief  can  be  afforded  till  the 
end  of  8  weeks,  what  must  be  the  daily  allowance  to  each 
man? 

51.  A  reservoir  has  3  pipes ;  the  first  can  fill  it  in  10  days, 
the  second,  in  16  days,  and  the  third  can  empty  it  in  20  days  : 
in  what  tune  will  the  cistern  be  filled  if  they  are  all  allowed 
to  run  at  the  same  time? 

58.  Two  persons,  A  and  B,  are  on  opposite  sides  of  a 
wood,    which   is    536    yards    in    circumference ;    they   begin    to 


PROMISCUOUS    EXAMPLES.  3C3 

travel  in  the  same  direction  at  the  same  time  ;  A  goes  at  the 
rate  of  11  yards  a  minute,  and  B,  at  the  rate  of  34  yards  in 
3  minutes  :  how  many  times  will  B  go  round  the  wood  before 
he  overtakes  A  ? 

59.  Two  men  and  a  boy  were  engaged  to  do  a  piece  of 
work.  One  of  the  men  could  do  it  in  10  days,  the  other  in 
1 6  days,  and  the  boy  could  do  it  in  20  days  :  how  long  would 
t  take  them  together  to  do  the  work  ? 

60.  A  owes  B  $500,  of  which  $150  is  to  be  paid  in  3 
months,  $175  in  6  months,  and  the  remainder  in  8  months  : 
what  would  be  the  equated  time  for  the  payment  of  the  whole  ? 

61.  If  42  men,  in  270  days,  working  8 J  hours  a  day,  can 
build  a  wall  98f  feet  long,  7J  feet  high,  and  2^  feet  thick  ; 
in  how  many  days  can  63  men  build  a  wall  45J  feet  long, 
6/^  feet  high,  and  3|-  feet  thick,  working  11 J  hours  a  day? 

62.  After  one-third  part  of  a  cask  of  wine  had  leaked 
away,  21  gallons  were  drawn,  when  it  was  found  to  be  half 
full  :   how  much  did  the  cask  hold  ? 

63.  A  man  had  a  bond  and  mortgage  for  $2500,  dated 
July  1st,  1854.  Not  satisfied  with  7Vo  interest,  he  sold  the 
mortgage  for  its  nominal  value,  and  on  Sept.  1st,  1854,  pur- 
chased 10  shares  of  railroad  stock,  par  $100,  at  115.  On  Nov. 
1st,  he  bought  8  shares  more  of  the  same  stock,  at  98  ;  and  on 
April  1st,  1855,  he  bought  5  shares  more  at  the  same  rate. 
On  the  first  days  of  August  and  February,  in  each  year,  he 
received  a  regular  semi-annual  dividend  of  4  per  cent.,  and  at 
the  end  of  the  year  (January  1st,  1856)  sold  his  whole  stock 
at  99  :  did  he  lose  or  gain  by  the  investment  in  stocks,  and 
how  much  ? 

64.  A  landlord  being  asked  how  much   he  received  for  the 
cut   of  his   property,   answered,  that   after   deducting   9  cents 

from  each  dollar,  for  taxes  and  repairs,  there  remained 
^3014.30  :   what  was  the  amount  of  his  rents  ? 

65.  If   165  pounds  of  soap  cost  $16.50,  for  liow  much  will 


3(14:  ANALYSIS   AND 

it  be  necessary  to  sell   390   pounds,  in  order   to  gain   the  cost 
of  36  pounds  ? 

66.  What  is  the  height  of  a  wall  which  is  14 J  yards  in 
length,  and  ^q  of  a  yard  in  thickness,  and  which  cost  $406,  it 
having  been  paid  for  at  the  rate  of  $10  per  cubic  yard? 

67.  A  thief  escaping  from  an  oflScer,  has  40  miles  the  start 
and  travels  at  the  rate  of  5  miles  an  hour ;  the  officer  in  pur 
suit  travels  at  the  rate  of  1  miles  an  hour  :  how  far  must  he 
travel  before  he  overtakes  the  thief? 

68.  Two  families  bought  a  barrel  of  flour  together,  for 
which  they  paid  $8,  and  agreed  that  each  child  should  count 
half  as  much  as  a  grown  person.  In  one  family  there  were  3 
grown  persons  and  3  children,  and  in  the  other,  4  grown  per- 
sons and  10  children  ;  the  first  family  used  from  the  flour  2 
weeks,  and  the  second  3  weeks  :  how  much  ought  each  to 
pay? 

69.  At  842  a  thousand,  how  many  thousand  feet  of  lumber 
should  be  given  for  a  farm  containing  33  A.  2R.  16  P.,  valued 
at  1125  an  acre? 

10.  A  person  paid  $150  for  an  insurance  on  goods,  at 
3|-  per  cent.,  and  finds  that  in  case  the  goods  are  lost,  he  will 
receive  the  value  of  the  goods,  the  premium  of  insurance,  and 
$25  besides  :   what  was  the  value  of  the  goods  ? 

n.  A  distiller  purchased  5000  bushels  of  rye,  which  he 
could  have  at  96  cents  a  bushel,  cash,  or  at  $1,  2  months' 
credit ;  which  would  be  the  more  advantageous,  to  buy  on 
credit,  or  to  borrow  the  money  at  7  per  cent.,  and  pay  the 
cash? 

12.  A  stockholder  bought  |  of  the  capital  of  a  company  at 
par  ;  he  sold  i  of  his  purchase  at  par,  and  the  remainder  for 
$25000,  and  by  the  latter  sale  made  $5000:  what  was  the 
value  of  the  whole  capital? 

13.  How  many  bushels  of  grain  will  a  bin  contain,  that  is 
3ft.  5 in.  wide,  2ft.  6 in.  long,  and  6ft.  deep? 


PROMISCUOUS   EXAMPLES.  365 

74.  Three  travelers  have  each  to  make  the  same  journey 
of  2160  miles  ;  the  first  travels  30  miles  a  day,  the  second 
27,  and  the  third  24  :  how  many  days  should  one  set  out 
after  the  other,  that  they  may  all  arrive  together? 

75.  A  house  which  was  resold  for  $7180,  would  have  given 
profit   of  $420,    if  the   second   proprietor   had  purchased   it 

^130  cheaper  than  he  did:  at  what  price  did  he  purchase  it? 

76.  A  piece  of  land  of  188  acres  was  cleared  by  two  com- 
panies of  men,  working  together  ;  the  first  numbered  25  men, 
and  the  second  22  ;  the  first  company  received  $84  more  than 
the  second  :  how  many  acres  did  each  company  clear,  and  what 
did  the  clearing  cost  per  acre  ? 

77.  I  have  three  notes  payable  as  follow :  one  for  $100, 
due  Feb.  12th;  the  second  for  $400,  due  March  12th;  and 
the  third  for  $300,  due  April  1st :  what  is  the  average  time 
of  payment  from  January  1st? 

78.  How  many  marble  slabs,  15  in.  square,  will  it  take  to 
pave  a  floor  32  feet  long,  and  25  feet  wide?  What  will  be 
tlie  cost  at  $3  a  square  yard  for  the  marble,  and  40  cents  a 
square  yard  for  labor? 

79.  A  man,  in  his  will,  bequeathed  $500  to  A,  $425  to  B, 
$300  to  C,  $250  to  D,  and  $175  to  E  ;  but  after  settling  up 
tlie  estate  and  paying  expenses,  there  was  but  $1155  left: 
what  is  each  one's  share  ? 

80.  If  31b.  of  tea  are  worth  71b.  of  coflfee,  and  1411).  of 
coffee  are  worth  481b.  of  sugar,  and  181b.  of  sugar  are  worth 
27  lb.  of  soap ;  how  many  pounds  of  soap  are  6  lb.  of  tea 
worth  ? 

81.  What  is  the  hour,  when  the  time  past  noon  is  |  the 
time  to  midnight  ? 

82.  If  f  of  a  yard  of  cloth  cost  $f ,  being  J  of  a  yard  wide, 
what  is  the  value  of  |-  of  a  yard  IJ  yards  wide,  of  the  same 
quality  ? 

83.  A  farmer  sold    60   fowls,    a   part    turkeys,    and    a  part 


S66  ANALYSi;^   AND 

chickens  ;  for  the  turkeys  he  received  $1.10  apiece,  and  for  the 
chickens  50  cents  apiece,  and  for  the  whole  he  received  $51  60 : 
how  many  were  there  of  each? 

84.  A  person  hired  a  man  and  two  boys  ;  to  the  man  he 
gave  6  shiUings  a  day,  to  one  boy  4  shilhngs,  and  to  the 
other  3  shiUings  a  day,  and  at  the  end  of  the  time  he  paid 
them  104  shillings  :  how  long  did  they  work? 

85.  Divide  $6471  among  three  persons,  so  that  as  often  as 
the  first  gets  |5,  the  second  will  get  $6,  and  the  third  $7. 

86.  Two  partners  have  invested  in  trade  11600,  by  which 
they  have  gained  1300  ;  the  gain  and  stock  of  the  second 
amount  to  $1140  :  what  is  the  stock  and  the  gain  of  each? 

87.  What  is  the  height  of  a  tower  that  casts  a  shadow 
75.75  yards  long,  at  the  same  time  that  a  perpendicular  staff 
3  feet  high,  gives  a  shade  of  4.55  feet  in  length? 

88.  A  can  do  a  certain  piece  of  work  in  3  weeks  ;  B  can 
do  3  times  as  much  in  8  weeks;  and  C  can  do  5  times  as 
much  in  12  weeks  :  in  what  time  can  they  all  together  do  the 
first  piece  of  work? 

89.  Two  persons  pass  a  certain  point,  at  an  interval  of  4 
hours ;  the  first  traveUng  at  the  rate  of  1 1  J,  and  the  second 
17  J  miles  an  hour  :  how  long,  after  passmg  the  fixed  point,  and 
how  far,  will  the  first  travel  before  he  is  overtaken  by  the  second  ? 

90.  Three  persons  engage  in  trade,  and  the  sum  of  their 
stock  is  $1600.  A^s  stock  was  in  trade  6  months,  B's  12 
months,  and  C's  15  months  ;  at  the  time  of  settlement,  A  re- 
ceives $120  of  the  gain,  B  $400,  and  C  $100  :  what  was  each 
person's  stock  ? 

91.  A,  B,  and  C,  start  at  the  same  time,  from  the  same 
point,  and  travel  in  the  same  direction,  around  an  island  73 
miles  in  circumference.  A  goes  at  the  rate  of  6  miles,  B  10 
miles,  and  C  16  miles  per  day  :  in  what  time  will  they  all  be 
together  again? 


PliOMISCUOUS   EXAMPLES.  367 

92.  What  length  of  wire,  J  of  an  inch  in  diameter,  can  be 
drawn  from  a  cube  of  copper,  of  2  feet  on  a  side,  allowing  10 
per  cent,  for  Waste? 

93.  A  person  having  $10000  invested  in  6  per  cent,  stocks, 
sells  out  at  65,  and  invests  the  proceeds  in  5  per  cents  at 
82  J  :  what  will  be  the  difference  in  his  annual  income  ? 

94.  In  order  to  take  a  boat  through  a  lock  from  a  certain 
river  into  a  canal,  as  well  as  to  descend  from  the  canal  into 
the  river,  a  volume  of  water  is  necessary  46 J  yards  long,  8 
yards  wide,  and  2f  yards  deep :  how  many  cubic  yards  of 
water  will  this  canal  throw  into  the  river  in  a  common  year, 
if  40  boats  ascend  and  40  descend  each  day,  except  Sundays 
and  eight  holidays  ? 

95.  A  company  numbering  sixty-six  shareholders  have  coa- 
Btructed  a  bridge  which  cost  $200000  :  what  will  be  the  gain 
of  each  partner  at  the  end  of  22  years,  supposing  that  G400 
persons  pass  each  day,  and  that  each  pays  one  cent  toll,  the 
expense  for  repairs,  &c.,  being  $5  per  year  for  each  share- 
holder? 

96.  Five  merchants  were  in  partnership  for  four  years,  the 
first  put  in  $60,  then,  5  months  after,  $800  ;  the  second  put 
in  first  $600,  and  6  months  after  $1800  ;  the  third  put  in 
$400,  and  every  six  months  after,  he  added  $500  ;  the  fourth 
did  not  contribute  till  8  months  after  the  commencement  of 
the  partnership  ;  he  then  put  in  $900,  and  repeated  this  sum 
every  6  months;  the  fifth  put  in  no  capital,  but  kept  the  ac- 
counts, for  which  the  others  agreed  to  allow  him  $800  a  year, 
to  be  paid  in  advance,  and  put  in  as  capital.  What  is  each 
one's  share  of  the  gain,  which  was  $20,000  ? 

9t.  A  general,  arranging  his  army  in  the  form  of  a  square, 
found  that  he  had  44  men  remaining  :  but  by  increasing  each 
side  by  another  man,  he  wanted  49  to  fill  up  the  square  :  how 
many  men  had  he? 

98.   A,  B,  and  C,  are   to   share  $987   in   the  proportion  of 


368  ANALYSIS   AND 

i  i  and  J  resi3ectively  ;  but  by  the  death  of  C,  it  is  required 
to  divide  the  whole  sum  proportionally  between  the  other  two  : 
what  will  each  have? 

99.  A  lady  going  out  shopping,  spent  at  the  first  place  she 
stopped,  one-half  her  money,  and  half  a  dollar  more  ;  at  the 
icxt  place,  half  the  remainder,  and  half  a  dollar  more  ;  and  at 
he  next  place,  half  the  remainder,  and  half  a  dollar  more, 
Allien  she  found  that  she  had  but  three  dollars  left :  how  much 
had  she  when  she  started  ? 

100.  If  a  pipe  of  6  inches  discharges  a  certain  quantity  of 
fluid  in  4  hours,  in  what  time  will  4  pipes,  each  of  3  inches 
bore,  discharge  twice  that  quantity? 

101.  A  man  bought  12  horses,  agreeing  to  pay  $40  for  the 
first,  and  in  an  increasing  arithmetical  progression  for  the  rest, 
paying  $310  for  the  last :  what  was  the  difference  in  the  cost, 
and  what  did  he  pay  for  them  all  ? 

102.  A  bill  for  goods,  amounting  to  $15000,  is  to  be  paid 
for  in  three  equal  payments  without  interest  ;  the  first  in  4 
months,  the  second  in  6  months,  and  the  third  in  9  months, 
money  being  worth  t  per  cent. :  how  much  ready  money  ought 
to  pay  the  debt  ? 

103.  If  an  iron  bar  5  feet  long,  2 J  inches  broad,  and  If 
inches  thick,  weigh  45  pounds,  how  much  will  a  bar  of  the 
same  metal  weigh,  that  is  Y  feet  long,  3  inches  broad,  and  2^ 
inches  thick? 

104.  A  market  woman  bought  a  certain  number  of  eggs  at 
the  rate  of  4  for  3  cents,  and  sold  them  at  the  rate  of  5  for 
4  cents,  by  which  she  made  4  cents  :  what  did  she  pay  apiece 
for  the  eggs?  What  did  she  make  on  each  egg  sold?  How 
many  did  she  sell  to  gain  4  cents? 

105.  A  person  passed  i  of  his  life  in  childhood,  ^^  of  it  in 
youth,  5  years  more  than  -i  of  it  in  matrunony  ;  he  then  had 
a  son,  whom  he  survived  4  years,  and  who  reached  only  i  the 
age  of  his  father  :   at  what  age  did  he  die  ? 


PROMISCUOUS  EXAMPLES.  369 

106.  A  well  is  to  be  stoned,  of  which  the  diameter  is  6 
feet  6  inches,  the  thickness  of  the  wall  is  to  be  1  foot  6 
inches,  leaving  the  diameter  of  the  well  within  the  wall  3  feet 
6  inches  ;  if  the  well  is  40  feet  deep,  how  many  cubic  feet  of 
stone  will  be  required? 

101.   A  surveyor  measured  a  piece  of  ground  in  the  form  of 
rectangle,  and  found  one  side  to  be  37  chains,  and  the  other 
■12  chains  16  links :   how  many  acres  did  it  contain  ? 

108.  A  farmer  bought  a  piece  of  land  for  $1500,  and 
agreed  to  pay  principal  and  interest  in  5  equal  annual  instal- 
ments :  if  the  interest  was  7  per  cent.,  how  much  was  the 
annual  payment  ? 

109.  A  fountain  has  4  receiving  pipes,  A,  B,  C,  and  D  ; 
A,  B,  and  C  will  fill  it  in  6  hours  ;  B,  C,  and  D  in  8  hours  ; 
C,  D,  and  A  in  10  hours  ;  and  D,  A,  and  B  in  12  hours  :  it 
has  also  4  discharging  pipes,  E,  F,  G,  and  II  ;  E,  F,  and  Q 
will  empty  it  in  6  hours  ;  F,  Or,  and  H  in  5  hours  ;  G,  H, 
and  E  in  4  hours  ;  H,  E,  and  F  in  3  hours.  Suppose  the 
fountain  full  of  water,  and  all  the  pipes  open,  in  wliat  time 
would  it  be  emptied? 

110.  If  a  ball  2  inches  in  diameter  weighs  5  pounds,  what 
will  be  the  diameter  of  another  ball  of  the  same  material  that 
weighs  78.12p  pounds  ? 

111.  A  gives  B  his  bond  for  $5000,  dated  April  1st,  1861, 
payable  in  ten  equal  annual  instalments,  the  first  payment  of 
$500  to  be  made  April  1st,  1802.  Afterward,  A  agreed  to 
take  up  his  bond  on  the  1st  day  of  April,  1863.  He  was  to 
pay,  on  that  day,  the  instalment  due  on  the  1st  of  April, 
1862,  with  interest  at  7  per  cent.,  the  instalment  due  April 
1st,  1863,  and  to  be  allowed  compound  interest,  at  7  per 
cent.,  to  be  computed  half-yearly,  on  each  of  the  subsequent 
payments:  what  sum,  on  the  first  day  of  April,  1863,  will 
cancel  the  bond? 


370 


MENSURATION. 


MENSURATION. 

405.  Mensuration  is  the  art  of  measuring,  and  embraces  all 
the  methods  of  determining  the  contents  of  geometrical  figures. 
It  is  divided  into  two  parts,  the  Mensuration  of  Surfaces,  and 
the  Mensuration  of  Volumes. 


o 
1—1 


MENSUKATION    OF  SURFACFS. 

406.  Surfaces  have  length  and  breadth.     They  are  measured 
by  means  of  a  square,  which  is  called  the  unit  of  surface. 

A  SQUARE  is  the  space  included  between  four  i  foot, 

equal  lines,  drawn  perpendicular  to  each  other. 
Each  line  is  called  a  side  of  the  square.  If 
each  side  be  one  foot,  the  figure  is  called  a 
square  foot. 

The  number  of  small  squares  that  is  con- 
tained in  any  large  square,  is  always  equal  to 
the  product  of  two  of  the  sides  of  the  large 
square.  As  in  the  figure,  3x3  =  9  square 
feet.  The  number  of  square  inches  contained 
in  a  square  foot  is  equal  to  12  X  12  =  144. 

If  the   sides    of  a   square  be  each  four  feet,  the  square  will 

contain  sixteen  square  feet.     For,  in  the  large  square  there  are 

sixteen   small   squares,  the    sides   of  which   are   each   one  foot. 

Therefore,  the  square  whose   side   is   four  feet,  contains  sixteen 

square  feet. 

Triangle. 

407.  A  TRIANGLE  is  a  figure  bounded  by  three  straight  lines. 
Thus,  ACB  is  a  triangle. 

The  lines  BA,  AC,  BC,  are  called  sides; 
and  the  corners,  B,  A,  and  C,  are  called 
a7igles.     The  side  AB  is  the  base. 

When  a  line  Hke  CD  is  drawn,  making 
the  angle  CD  A  equal  to  the  angle  CDB, 
then  CD  is  said  to  be  at  right   angles  to 


OF   SURFACES.  37] 

AB,  and  CD  is  called  the  allitade  of  the  triangle.  Each  tri- 
angle CAD  or  CDB  is  called  a  right-angled  triangle.  The 
side  BC,  or  the  side  AC,  opposite  the  right  angle,  is  called 
the  hypothenuse. 

The   area   or  contents   of  a   triangle  is  equal   to   half  the 
product   of  its  base  by  its  aUitude  (Bk.  lY.,  Pro2?.  YL). 

Note. — All  the  references  are  to  Davies'  Legendre. 

Examples. 

1.  The  base,  AB,   of  a  triangle  is   50   yards,  and  the  per- 
pendicular, CD,  30  yards  :  what  is 

the  area?  operation. 

60 
Analysis. — Wo    first    multiply  tho  30 

base  by  the  altitude,  and  the  product  2)1500 

is  square  yards,  which  we  divide  by      ^^^^^    "^^  ^^^^ 

2  for  the  area. 

2.  In  a  triangular  field  the  base  is  60  chams,  and   the  per- 
pendicular 12  chains  :  how  mjich  does  it  contain  ? 

3.  There  is  a  triangular  field,  of  which  the  base  is  45  rods, 
and  the  perpendicular  38  rods  :  what  are  its  contents  ? 

4.  What   are    the   contents  of  a  triangle  whose   base   is  15 
chains,  and  perpendicular  36  chains? 

Rectangle  and  Parallelogram. 

408.  A  RECTANGLE  is  a  four-sided  figure, 
or  quadrilateral,  like  a  square,  in  which  the 
sides  are  perpendicular  to  each  other,  but 
the  adjacent  sides  are  not  equal. 

409.  A  PARALLELOGRAM  is  a  quadrilat- 
eral which  has  its  opposite  sides  equal 
and  parallel,  but  its  angles  not  right 
angles.  The  line  DE,  perpendicular  to  the 
base,  is  called  the  altitude. 


372  MENSURATION 

Tlie  area  of  a  square,  rectangle,  or  iJarallelogram,  is  equal 
to  the  x^roduct  of  the  base  and  altitude. 

Examples. 

1.  What  is  the  area  of  a  square  field,  of  which  the  sides 
are  each  66.16  chains? 

2.  What  is  the  area  of  a  square  piece  of  land,  of  which  the 
ides  are  54  chains  ? 

3.  What  is  the  area  of  a  square  piece  of  land,  of  which  tlie 
sides  are  75  rods  each  ? 

4.  What  are  the  contents  of  a  rectangular  field,  the  length 
of  which  is  80  rods,  and  the  breadth  40  rods  ? 

5.  What  are  the  contents  of  a  field  80  rods  square? 

6.  What  are  the  contents  of  a  rectangular  field,  30  chains 
long  and  5  chains  broad? 

T..  What  are  the  contents  of  a  field,  54  chains  long  and  18 
rods  broad? 

8.  The  base  of  a  parallelogram  is  542  yards,  and  the  per- 
pendicular height  120  feet :  what  'is  the  area  ? 

9.  The  measure  of  a  rectangular  field  is  24000  square  feet, 
and  its  length  is  200  feet :  what  is  its  breadth  ? 

Trapezoid. 

410.    A    TRAPEZOID    is   a   quadrilateral,  D 

ABCD,  having  two  of  its  opposite  sides,  / 

AB,     DO,    parallel.     The    pei-pendicular,        / 

EF,  is  called  the  altitude.  A            F           B 

The  area  of  a  trapezoid  is  equal  to  half  the  product  of  the 
sum  of  the  two  parallel  sides  by  the  altitude  {Bk.  lY.,  Prop. 
TIL). 

Examples. 
1.   Required   the   area  or   contents  of  the   trapezoid  ABCD, 
having     given    AB  =  643.02     feet,     DC  =  428.48    feet,     and 
EF  =  342.32  feet. 


OF  SURFACES.  373 

Analysis. — We  first  find  operation. 

tho    sum    of    the    paraUel'  643.02  +  428.48  =  1071.50  =  sum 

sides,  and  then  multiply  it  of  parallel  sides.     Then,  1071.50  x 

by  the  altitude ;  after  which  342.32  =  3GG795.88  •  and  ^^-ilii-^J  =: 

we   divide   the  product  by  183397.94  =  the  area. 
2,  for  the  area. 

2.  What  is  tlie  area  of  a  trapezoid,  the  parallel  sides  of 
which  are  24.82  and  16.44  chains,  and  the  perpendicular  dis- 
tance between  them  10.30  chains? 

3.  Requu-ed  the  area  of  a  trapezoid,  whose  parallel  sides 
are  51  feet  and  37  feet  6  inches,  and  the  perpendicular  dis- 
tance between  them  20  feet  and  10  inches. 

4.  Required  the  area  of  a  trapezoid,  whose  parallel  sides 
are  41  and  24.5,  and  the  perpendicular  distance  between  them 
21.5  yards. 

5.  What  is  the  area  of  a  trapezoid,  whose  parallel  sides  are 
15  chains,  and  24.5  chains,  and  the  perpendicular  height  30.80 
chains  ? 

C.  What  are  the  contents  of  a  trapezoid,  when  the  parallel 
sides  are  40  and  64  chains,  and  the  perpendicular  distance  be- 
tween them  52  chains? 

Circle. 

411.  A  Circle  is  a  portion  of  a  plane  bounded  by  a  curved 
li-ne,  every  point  of   which  is  equally  dis- 
tant  from   a   certain   point  within,  called 
the  center. 

The  curved  line  AEBD  is  called  the 
circumference ;  the  point  C,  the  center ; 
the  line  AB,  passing  through  the  center, 
a  diameter ;   and  CB,  a  radius. 

The    circumference,    AEBD,  is    3.1416 
times  as  great  as  the  diameter  AB.     Hence,  if  the  diameter  is 
1,  the  circumference  will  be  3.1416.     Therefore,  if  the  diameter 
is   known,  the  circumference  is  found  by  multiplying  3.1416 
by  the  diameter  {Bk.  V.,  Prop,  XVI.). 


374  MENSURATION 

Examples. 

1.  The  diameter  of  a  circle  is  8  :  what  is  the  circumference? 

OPERATION. 

Analysis. — The   circumference  is  found  3.1416 

by   simply   multiplying   3.1416   by  the   di-  8 

^'^^t^^-  Ans.  25.1328  i 

2.  The  diameter  of  a  circle  is  186  :  what  is  the  circum- 
ference ? 

3.  The  diameter  of  a  circle  is  40  :  what  is  the  circum- 
ference ? 

4.  What  is  the  circumference  of  a  circle  whose  diameter  is 

5t? 

412.  Since  the  circumference  of  a  circle  is  3.1416  times  as 
great  as  the  diameter,  it  follows,  that  if  the  circumference  is 
knoivn,  we  may  find  the  diameter  by  dividing  it  by  3.1416. 

Examples. 

1.  What   is   the  diameter  of  a  circle  whose  circumference  is 

157.08? 

2.  What  is  the  diameter  of  a  circle  whose  circumference  is 

23304.3888? 

3.  What  is  the  diameter  of  a  circle  whose  circumference  is 
13700? 

413.    To  find  the  area  or   contents   of  a   circle. 

Hule.— Multiply  the  square  of  the  radius  by  3.1416  {Bk. 
Y.,  Prop.  XY.). 

Examples. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  12  ? 

2.  What  is  the  area  of  circle  whose  diameter  is  5  ? 

3.  What  is  the  area  of  a  circle  whose  diameter  is  14? 


OF   VOLUMES. 


375 


4.  now  many  square  yards  iu  a  circle  whose  diameter  is  3J 
feet  ? 

5.  What   is  the  area  of  a  circle  whose  dianiotcr  is   -J   mile  ? 


Sphere. . 

414.  A  SPHERE  is  a  portion  of  space 
bounded  by  a  curved  surface,  all  the 
l)oints  of  which  are  equally  distant  from 
a  certain  point  within,  called  the  center. 
The  line  AD,  passing  through  its  center 
C,  is  called  the  diameter  of  the  sphere, 
and  AC  its  radius. 


415.    To  find  the  surface  of  a  sphere. 

Rule. — 3TuUipIy  (he  square  of  the  diameter  by  3.1416  {Bh 
VIII.,  Prop.  X.,  Cor.  1.). 

Examples. 

1.  What  is  the  surface  of  a  sphere  whose  diameter  is  6? 

2.  What  is  the  surface  of  a  sphere  whose  diameter  is  14  ? 

3.  Required  the  number  of  square  inches  in  the  surface  of  a 
sphere  whose  diameter  is  3  feet  or  36  inches. 

4.  Required  the  area  of  the  surface  of  the  earth,  its  mean 
diameter  being  7918.7  miles? 

MENSUKATION    OF    VOLUMES. 
416.    A  SOLID  or  volume  is  a  portion  of  space   having  three 
dimensions :  length,  breadth,  and 
thickness.     It   is  measured  by  a 
cube,  called   the   cubic  unit,   or 
unit  of  volume. 

A    CUBE    is   a  volume    having 

six  equal  faces,  which  are  squares. 

If  the  sides  of  the  cube  be  each  o  ,-   i.      i        i 

0  lect  =  1  yard. 

one  foot  long,  the  figure  is  called 

a  cubic  foot    But  when  the  sides  of  the  cube   arc   one  yard, 


1 

^-y  /  A 

in 

':':P 

■r-< 

Pi 

II 

■■'1 

■-^ 

:                     !               '¥ 

:     I           1           f 

376 


MENSURATION 


as  in  the  figure,  it  is  called  a  cubic  yard.  The  base  of  the 
cube,  which  is  the  face  on  which  it  stands,  contains  3x3  =  9 
square  feet.  Therefore,  9  cubes,  of  one  foot  each,  can  be 
placed  on  the  base.  If  the  figure  were  one  foot  high,  it  would 
contain  9  cubic  feet ;  if  it  were  2  feet  high,  it  would  contain 
two  tiers  of  cubes,  or  18  cubic  feet ;  hence,  the  contents  are 
equal  to  the  product  of  the  length,  breadth,  and  height. 

417.     To  find  the  volume  or  contents  of  a  sphere. 

Rule. — Multiply  the  surface  by  the  diameter,  and  divide 

the  product  by  6  ;  the  quotient  will  be  the  contents  {Bk  VIII., 

Prop.  XIY.,  Sch.  3), 

Examples. 

1.   What  are  the  contents  of  a  sphere  whose  diameter  is  12  ? 


Analysis. — We  find  the  sur- 
face by  multiplying  the  square  of 
the  diameter  by  3.1416.  We 
then  multiply  the  surface  by  the 
diameter,  and  divide  the  product 
by  6. 


OPERATION. 

12'  =  144 
multiply  by         3.1416 

surface  452.3904 

diameter  12 

6)5428.6848 


solidity 


904.7808 


2.  What  are  the  contents  of  a  sphere  whose  diameter  is  8  ? 

3.  Find  the  contents  of  a  sphere  whose  diameter  is  16  inches. 

4.  What   are  the  contents   of  the  earth,  its   mean   diameter 
being  7918.7  miles? 

5.  Find  the  contents  of  a  sphere  whose  diameter  is  1.2  feet. 


Prism. 
418.   A  Prism  is  a  volume  whose  ends  or  bases 
are  equal  plane  figures,  and   whose  faces   are   par- 


allelograms. 


The  sum  of  the  sides  which  bound  the  base,  is 
called  the  perimeter  of  the  base  ;  and  the  sum  of 
the  parallelograms  which  bound  the  prism,  is  called 
the  eonvcx  surface. 


(CD 


OF   VOLUMES.  377 

419.    To  find  the  convex  surface  of  a  right  prism. 

Rule. — Multiply  the  perimeter  of  the  base  by  the  perpen- 
dicular  height,  and  the  product  will  be  the  convex  surface 
(Bk.  YIT.,  Prop).  I.). 

Examples. 

1.  What  is  the  convex  surface  of  a  prism  whose  base  is 
bounded  by  five  equal  sides,  each  of  which  is  35  feet,  the  alti- 
tude being  52  feet? 

2.  What  is  the  convex  surface  when  there  are  eight  equal 
sides,  each  15  feet  in  length,  and  the  altitude  is  12  feet? 

420.    To  find  the  volume  or  contents  of  a  prism. 

Rule. — Multiply  the  area  of  the  base  by  the  altitude,  and 
the  product  ivill  be  the  contents   {Bk.  YII.,  Prop.  XIV.). 

Examples. 

1.  What  are  the  contents  of  a  square  prism,  each  side  of 
the  square  which  forms  the  base  being  16,  and  the  altitude  of 
the  prism  30  feet? 

OPERATION. 

Analysis. — We   first  find   the   area  of   the         Tp2  _  or/* 
square  which  forms  the  base,  and  then  multi-  ~      on 


ply  by  the  altitude.  . 


1680 


2.  What  are  the  contents  of  a  cube,  each  side  of  which  is 
48  inches? 

3.  IIow  many  cubic  feet  in  a  block  of  marble,  of  whicl 
the  length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and 
height  or  thickness  5  feet  ? 

4.  How  many  gallons  of  water  will  a  cistern  contain,  whose 
dimensions  are  the  same  as  in  the  last  example? 

5.  Required  the  measure  of  a  triangular  prism,  whose  height 
is  20  feet,  and  area  of  the  base  691. 


378 


MENSURATI02J' 


Cylinder. 

421.  A  Cylinder  is  a  volume  generated 
by  the  revolution  of  a  rectangle,  AF,  about 
BF.  The  line  EF  is  called  the  axis,  or 
altitude ;  the  circular  surface,  the  convex 
surface  of  the  cylinder ;  and  the  circular 
eads,  the  hasea. 

422.  To  find  the  convex  surface  of  a 
cylinder. 

Rule. — Multiply    the    circumference  of   the    base    by    the 

altitude,  and  the  product  will  be  the  convex  surface    {Book 

Yin.,  Prop,  I.). 

Examples. 

1.  What   is   the  convex   surface  of  a  cylinder,  the  diameter 
of  whose  base  is  20,  and  the  altitude  40  ? 

2.  What   is  the  convex  surface  of  a  cylinder  whose  altitude 
is  28  feet,  and  the  circumference  of  its  base  8  feet  4  inches  ? 

3.  What   is   the   convex   surface   of  a  cylinder,  the  diameter 
of  whose  base  is  15  inches,  and  altitude  5  feet  ? 

4.  What   is   the   convex   surface  of  a  cylinder,  the  diameter 
of  whose  base  is  40,  and  altitude  50  feet  ? 

423.    To  find  the  volume  or  contents  of  a  cylinder. 

Rule. — 3fuUiply  the  area  of  the  base  by  the  altitude:   the 
product   will   be  the  contents   or  volume   (Book  YIII.,  Prop. 

II.). 

Examples. 
1.   Required  the  contents  of  a  cylinder  of  which  the  altitude 
is  1 1  feet,  and  the  diameter  of  tlie 
base  16  feet. 


Analysis. — We  first  find  the  area 
of  the  base,  and  then  multiply  by 
the  altitude:  the  product  is  the  so- 
lidity. 


OPERATION. 

16^=256 

.1854 


area  base,  201.0624 
11 


2211.6864 


OF  VOLUMES.  379 

2.  What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  40,  and  the  altitude  29  ? 

3.  What   are   the   contents   of  a   cylinder,   the  diameter   of 
whose  base  is  24,  and  the  altitude  30  ? 

4.  What   are  the   contents    of   a   cylinder,  the   diameter   of 
whose  base  is  32,  and  altitude  12  ? 

5.  What    are    the   contents   of   a   cylinder,  the   diameter  o 
whose  base  is  25  feet,  and  altitude  15  ? 

Pyramid. 

424.  A  Pyramid  is  a  volume  bounded 
by  several  triangular  planes  united  at 
the  same  point,  S,  called  the  vertex,  and 
by  a  plane  figure  or  base,  ABODE,  in 
which  they  terminate.  The  altitude  of 
tlie  pyramid  is  the  line  SO,  drawn  per- 
pendicular to  the  base. 

A 
425.    To  find  the  volume  or  contents  of  a  pyramid. 

Rule. — Multiply  the  area    of  the  base  by  the  altitude^  and 
divide  the  product  by  3    {Bk,  VII.,  Prop.  XVII.). 

Examples. 

1.  Required  the  contents  of  a  pyramid,  operation. 
the  area  of  whose  base  is  86,  and  the  alti-  86 
tude  24.  ^^ 

Analysis. — We  simply  multiply  the  area  of  ll 

the  base  86,  by  the  altitude   24,  and  then  di-         ^ns.    6SS 
vide  the  product  by  3. 

2.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
ase  is  365,  and  the  altitude  36? 

3.  What   are  the  contents  of  a  pyramid,  the  area  of  whoso 
base  is  207,  and  altitude  36? 

4.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
biise  is  562,  and  altitude  30  ? 


380 


MENSUKATION     OF   VOLUMES. 


5.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
base  is  540,  and  altitude  32  ? 

6.  A  pyramid  has  a  rectangular  base,  the  sides  of  which  are 
50  and  24  ;  the  altitude  of  the  pyramid  is  36  :  what  are  its 
contents  ? 

7.  A  pyramid  with  a  square  base,  of  which  each  side  is  15 
has  an  altitude  of  24  :   what  are  its  contents  ? 

Cone. 

426.  A  Cone  is  a  volume  generated  by 
the  revolution  of  a  right-angled  triangle 
ASC,  about  the  side  CB.  The  point  C  is 
the  vertex,  and  the  line  CB  is  called  the 
axis,  or  altitude. 

427.  To  find  the  volume  or  contents  of  a 
cone. 

Rule. — 3Iultiply  the  area  of  the  base  by  the  altitude,  and 
divide  the  product  by  3  ;  or,  multiply  the  area  of  the  base 
by  one-third  of  the  altitude   {Bk.  YIII.,  Prop.  Y.). 

Examples. 
1.   Required  the  contents   of  a   cone,  the  diameter  of  whose 
base  is  6,  and  the  altitude  11. 


Analysis. — We  first  square  the 
diameter,  and  multiply  it  by 
.7854,  which  gives  the  area  of 
the  base.  We  next  multiply  by 
the  altitude,  and  then  divide  the 
product  by  3. 


OPERATION. 

6^^  =  36 

36  X  .1854   =  28.2744 
11 


3)311.0184 
Ans.     103.6728 


2.  What  are  the  contents  of  a  cone,  the  diameter  of  whose 
base  is  36,  and  the  altitude  27  ? 

3.  What  are  the  contents  of  a  cone,  the  diameter  of  whose 
base  is  35,  and  the  altitude  27  ? 

4.  What   are   the  contents  of  a  cone,  whose   altitude   is   27 
feet,  and  the  diameter  of  the  base  20  feet? 


GAUGING. 


381 


aAUG-ING. 

428.  Cask-Gauging  is  the  method  of  finding  the  number  of 
gallons  which  a  cask  contains,  by  measuring  the  external  di- 
mensions of  the  cask. 

429.  Casks  are  divided  into  four  varieties,  according  to  the 
L'urvature  of  their  sides.  To  which  of  the  varieties  any  cask 
belongs,  must  be  judged  of  by  inspection. 


1st   Variety — least  curvature. 


2d    Variety — least  mean  curvature. 


3d    Variety — greatest  mean  curvature. 


4th  Variety — greatest  curvature. 


430.  The  first  thing  to  be  done  is  to  find  the  mean  di- 
ameter.    To  do  this, 

Rule. — Divide  the  head  diameter  by  the  bung  diameter, 
and  find  the  quotient  in  the  first  column  of  the  following 
table,  marked  Qu.  Then  if  the  bung  diameter  be  multijjlied 
by  the  number  on  the  same  line  with  it,  and  in  the  column 
answering  to  the  proper  variety,  the  product  will  be  the  true 
mean  diameter,  or  the  diameter  of  a  cylinder  having  the 
same  altilude  and  the  same  contents  with  the  cask  proposed. 


oSl 


GAUGING. 


50 

l8t  Var. 

2(1  Var. 

3d  Vmr. 

4th  Var. 

Qu. 

Ist  Var. 

2d  Var. 

3d  Var. 

4tli  Var 

8660 

8465 

7905 

7637 

76 

9370 

9337 

8881 

8827 

51 

8680 

8493 

7937 

7681 

77 

9396 

9358 

8944 

8874 

52 

8700 

8520 

7970 

7725 

78 

9334 

9290 

8967 

8922 

53 

8720 

8548 

8002 

7769 

79 

9353 

9320 

9011 

8970 

54 

8740 

8576 

8036 

7813 

80 

9380 

9353 

9055 

9018 

55 

8700 

8605 

8070 

7858 

81 

9409 

9383 

9100 

9066 

5G 

8781 

8633 

8104 

7903 

83 

9438 

9415 

9144 

9114 

57 

8803 

8662 

8140 

7947 

83 

9467 

9446 

9189 

9163 

58 

8834 

8690 

8174 

7993 

84 

9496 

9478 

9334 

9211 

59 

8846 

8720 

8210 

8037 

85 

9536 

9510 

9280 

9260 

60 

8869 

8748 

8246 

8083 

86 

9556 

9543 

9336 

9308 

61 

8893 

8777 

8283 

8138 

87 

9586 

9574 

9372 

9357 

63 

8915 

8806 

8330 

8173 

88 

9616 

9606 

9419 

9406 

63 

8938 

8835 

8357 

8330 

89 

9647 

9638 

9466 

9455 

64 

8963 

8865 

8395 

8365 

90 

9678 

9671 

9513 

9504 

65 

8986 

8894 

8433 

8311 

91 

9710 

9703 

9560 

9553 

66 

9010 

8924 

8473 

8357 

93 

9740 

9736 

9608 

9602 

67 

9034 

8954 

8511 

8404 

93 

9773 

9768 

9656 

9652 

68 

9060 

8983 

8551 

8450 

94 

9804 

9801 

9704 

9701 

69 

9084 

9013 

8590 

8497 

95 

9836 

9834 

9753 

9751 

70 

9110 

9044 

8631 

8544 

96 

9868 

9867 

9802 

9800 

71 

9136 

9074 

8673 

8590 

97 

9901 

9900 

9851 

9850 

73 

9163 

9104 

8713 

8637 

98 

9933 

9933 

9900 

9900 

73 

9188 

9135 

8754 

8685 

99 

9966 

9966 

9950 

9950 

74 

9215 

9166 

8796 

8733 

100 

10000 

10000 

10000 

10000 

75 

9243 

9196 

8838 

8780 

Examples. 

1.  Supposing  the  diameters  to  be  32  and  24,  it  is  required 
to  find  the  mean  diameter  for  each  variety. 

Dividing  24  by  32,  we  obtain  .75 ;  which  being  found  in 
the  column  of  quotients,  opposite  thereto  stand  the  numbers, 


.9242 
.9196 


which  being  each  mul- 
tiplied   by  32,    produce 


I  8780  J  ^^sP^^^t^^'^ly, 


29.5U4 
29.4272 
28.2816 
28.0960 


for  the  correspond- 
ing mean  dianietca 
required. 


2.  The  head  diameter  of  a  cask  is  26  inches,  and  the  bung 
diameter  3  feet  2  inches :  what  is  the  mean  diameter,  the  cask 
being  of  the  third  variety  ? 

3.  The  head  diameter   is   22   inches,  the   bung   diameter   34 


GAl'GINQ.  S83 

iiiclies  :    what   is   the   mean  diameter  of  a  cask   of  the  fourth 
variety  ? 

431.  Having  found  the  mean  diameter,  we  multiply  the 
square  of  the  mean  diameter  by  the  decunal  .7854,  and  tlie 
product  by  the  length  ;  this  will  give  the  contents  in  cubic 
inches.  Then,  if  we  divide  by  231,  we  have  the  contents  iii 
wine  gallons  (see  Art.  4*75)  ;  or  if  we  divide  by  282,  we  have 
the  contents  in  beer  gallons  (Art.  4^6). 

Analysis. — For    wine    measure,    we  operation. 

multij)ly   the   length   by  the   square   of         I  X  d*  X  ^-^^  = 
the   mean  diameter,  then   by  the  deci-         I  x  (P  X   0034 
mal  .7854,  and  divide  by  231. 

If,  then,  we  divide  the  decimal  .7854  by  231,  the  quotient  car- 
ried to  four  places  of  decimals  is  .0034 ;  and  this  decimal,  multiplied 
by  the  square  of  the  mean  diameter  and  by  the  length  of  the 
cask,  will  give  the  contents  in  wine  gallons. 

For   similar  reasons,   the  content  is  operation. 

found    in   beer   gallons   by  multiplying         ;        72  ^  7854  

together  the   length,  the  square  of  the 

mean  diameter,  and  the  decimal  .0028.         I  X  d^  X  .0028. 

Hence,  for  gauging  or  measuring  casks, 

Tlule.— Multiply  the  length  by  the  square  of  the  mean  di 
ameter ;  then  multiply  by  34  for  ivine,  and  by  28  for  beer 
measure,  and  point  off  in  the  product  four  decimal  places. 
The  jyroduct  will  then  express  gallons,  and  the  decimals  of  a 
gallon. 

1.  How  many  wine  gallons  in  a  cask,  whose  bmig  diameter 
is  30  inches,  head  diameter  30  inches,  and  length  50  inches  ; 
the  cask  being  of  the  first  variety? 

2.  How  many  wine,  and  how  many  beer  gallons  in  a  cask 
\\'hose  length  is  36  inches,  bung  diameter  35  inches,  ^nd  head 
diameter  30  inches,  it  being  of  the  first  variety? 

3.  How  many  wine  gallons  in  a  cask  of  which  the  head 
diameter  is  24  inches,  bung  diameter  36  inches,  and  length  3 
feet  6  inches,  the  cask  being  of  the  second  variety? 


384:  MECHANICAL    POWERS. 


OF    THE    MECHANICAL    POWERS. 

432.  There  are  six  simple  machines,  which  are  called  Me- 
chanical powers.  They  are,  the  Lever,  the  Pulley,  the  Wheel 
and  Axle,  the  Inclined  Plane,  the   Wedge,  and  the  Screw. 

433.  To  understand  the  nature  of  a  machine,  four  things 
must  be  considered. 

1st.  The  power  or  force  which  acts.  This  consists  in  the 
efforts  of  men  or  horses,  of  weights,  springs,  steam,  &c. : 

2d.  The  resistance  which  is  to  be  overcome  by  the  power. 
This  generally  is  a  weight  to  be  moved : 

3d.  The  center  of  motion,  called  a  fulcrum  or  prop.  The 
prop  or  fulcrum  is  the  point  about  which  all  the  parts  of  the 
machine  move  : 

4th.   The  respective  velocities  of  the  power  and  resistance. 

434.  A  machine  is  said  to  be  in  equilibrium  when  the  re- 
sistance exactly  balances  the  power ;  in  which  case  all  the 
parts  of  the  machine  are  at  rest,  or  in  uniform  motion,  and  in 
the  same  direction. 

Lever. 

435.  The  Lever  is  a  bar  of  wood  or  metal,  which  moves 
around  the  fulcrum.     There  are  three  kinds  of  levers. 


1st.    When     the     fulcrum     is 
between     the    weight     and     the 


ower : 


2d.  When  the  weight  is 
between  the  power  and  the 
fulcrum  : 


MECHANICAL    POWERS.  385 


; 


3d.  When  the  power  is 
between  the  fulcrum  and  tlie 
weight :  t==== 

The   perpendicular   distance 
from    the   fulcrum  to  the  di- 
rections   of   the  weight   and  power,   are  called  the   arms   of 
the  lever.  , 

436.  An  equilibrium  is  produced  in  all  the  levers,  when  the 
weight,  multiplied  by  its  distance  from  the  fulcrum,  is  equal  to 
the  power  multiplied  by  its  distance  from  the  fulcrum.    That  is, 

Rule. — The  weight  is  to  the  power,  as  the  distance  from 
Jie  power  to  the  fulcrum^  is  to  the  distance  from  the  weight 
to  the  fulcrum. 

Examples. 

1.  In  a  lever  of  the  first  kind,  the  fulcrum  is  placed  at  the 
middle  point  :  what  power  will  be  necessary  to  balance  a 
weight  of  40  pounds  ? 

2.  In  a  lever  of  the  second  kind,  the  weight  is  placed  at 
the  middle  point :  what  power  will  be  necessary  to  sustain  a 
weight  of  50  lb.  ? 

3.  In  a  lever  of  the  third  kind,  the  power  is  placed  at  the 
middle  point  :  what  power  will  be  necessary  to  sustain  a 
weight  of  25  lb.  ? 

4.  A  lever  of  the  first  kind  is  8  feet  long,  and  a  weight  of 
CO  lb.  is  at  a  distance  of  2  feet  from  the  fulcrum :  what  power 
will  be  necessary  to  balance  it  ? 

5.  In  a  lever  of  the  first  kind,  that  is  6  feet  long,  a  weight 
of  200  lb.  is  placed  at  1  foot  from  the  fulcrmn :  what  power 
will  balance  it  ? 

6.  In  a  lever  of  the  first  kind,  hke  the  common  steelyard, 
the  distance  from  the  weight  to  the  fulcrum  is  one  inch  ;  at 
what   distance   from   the  fulcrum    must   the   poise    of   1  lb.    bo 

17 


SS6  MECHANICAL   POWEkS. 

placed,    to   balance   a  weight   of    lib?     A   weight   of   l|Ib.  ? 
Of  2  lb.  ?     Of  4  lb.  ? 

7.  In  a  lever  of  the  third  kind,  the  distance  from  the  ful- 
crum to  the  power  is  5  feet,  and  from  the  fulcrum  to  the 
weight  8  feet  :  what  power  is  necessary  to  sustain  a  weight  of 
40  lb.  ? 

8.  In  a  lever  of  the  third  kind,  the  distance  from  the  ful- 
crum to  the  weight  is  12  feet,  and  to  the  power  8  feet :  what 
power  will  be  necessary  to  sustain  a  weight  of  1001b.? 

437.  Remarks. — In  determining  the  equilibrium  of  the  lever, 
we  have  not  considered  its  weight.  In  levers  of  the  first  kind, 
the  weight  of  the  lever  generally  adds  to  the  power,  but  in 
the  second  and  third  kinds,  the  weight  goes  to  diminish  the 
effect  of  the  power. 

In  the  previous  examples,  we  have  stated  the  circumstances 
under  which  the  power  will  exactly  sustain  the  weight.  In 
order  that  the  power  may  overcome  the  resistance,  it  must  of 
course  be  somewhat  increased.  The  lever  is  a  very  important 
mechanical  power,  being  much  used,  and  entering,  indeed,  into 
most  other  machines. 

Of  the   Pulley. 

438.  The    pulley  is    a  wheel,  having  a         t         ,   ■         i 
groove   cut    in    its   circumference,   for   the  /^    ^v 
purpose  of  receiving   a  cord  which  passes 
over  it.     When  motion  is  imparted  to  the 
cord,  the  pulley  turns  around  its  axis,  which 
is    generally   supported  by  being   attached         ^^         ^J 
to  a  beam  above. 

439.  Pulleys  are  divided  into  two  kinds,  fixed  pulleys  and 
movable  pulleys.  When  the  pulley  is  fixed,  it  does  not  increase 
the  power  which  is  applied  to  raise  the  weight,  but  merely 
changes  the  direction  4n  which  it  acts. 


MECHANICAL    rOWKRS. 


3ST 


advantage.    Thus, 


r 


440.  A  movable  pulley  gives  a  mechanical 
iu  the  movable  pulley,  the  hand  which  sus- 
tains the  cask  actually  supports  but  one- 
half  of  the  weight  of  it ;  the  other  half  is 
supported  by  the  hook  to  which  the  other 
end  of  the  cord  is  attached. 

441.  If  we  have  several  movable  pul- 
leys, the  advantage  gained  is  still  greater, 
and  a  very  heavy  weight  may  be  raised  by 
a  small  power.  A  longer  time,  however, 
will  be  required,  than  with  the  single  pulley. 
It  is,  indeed,  a  general  principle  in  ma- 
cliines,  that  what  is  gained  in  power,  is 
lost  in  time;  and  this  is  true  for  all  ma- 
chines. There  is  also  an  actual  loss  of 
power,  viz.,  the  resistance  of  the  machine 
to  motion,  arising  from  the  rubbing  of  the 
parts  against  each  other,  which  is  called 
the  friction  of  the  machine.  This  varies  in 
the  different  machines,  but  must  always  be 
allowed  for,  in  calculating  the  power  neces- 
sary to  do  a  given  work.  It  would  be 
wrong,  however,  to  suppose  that  the  loss 
was  equivalent  to  the  gain,  and  that  no 
advantage  is  derived  from  the  mechanical 
powers.  We  are  unable  to  augment  our 
strength,  but  by  the  aid  of  science  we  so 
divide  the  resistance,  that  by  a  continued 
exertion    of    power,     we    accomplish    that 

which  it  would  be  impossible  to  effect  by  a  single  effort. 

If,  in  attaining  this  result,  we   sacrifice  time,  we  cannot  but 
see  that  it  is  most  advantageously  exchanged  for  power. 


442.   It  is  plain,  that  in  the  movable  pulley,  all  the  parts  of 
the  cord  will  be   equally  stretched  ;   and  hence,  each  cord  run- 


388  MECHANICAL    PuWERS. 

ning  from    pulley  to   pulley,    will   bear   an  equal   part   of  the 
weight ;  consequently, 

Rule. — The  power  will  always  he  equal  to  the  weight 
divided  by  the  number  of  cords  which  reach  from  pulley  to 
pulley. 

Examples. 

1.  In  a  single  immovable  pulley,  what  power  will  support 
weight  of  601b.? 

2.  In  a  single  movable  pulley,  what  power  will  support  a 
weight  of  80  lb.  ? 

3.  In  two  movable  pulleys,  with  4  cords  (see  last  fig.),  what 
power  will  support  a  weight  of  1001b.? 


443.  This  machine  is  com- 
posed of  a  wheel  or  crank, 
firmly  attached  to  a  cylin- 
drical axle.  The  axle  is 
supported  at  its  ends  by 
two  pivots,  which  are  of 
less  diameter  than  the  axle 
around  which  the  rope  is 
coiled,  and  which  turn  freely 
about  the  points  of  support. 
In  order  to  balance  the 
weight,  we  must  have, 

Rule. — The    power   to   the   weight,   as    the  radius  of  the 
axle,  to  the  length  of  the  crank,  or  radius  of  the  tvheel. 

Examples. 
1.    What  must  be   the  length  of    a   crank  or  radius   of    a 
wheel,  in   order  that   a  power   of  401b.  may  balance  a  weight 
of  600  lb.  suspended  from  an  axle  of  6  inches  radius  ? 


MECHANICAL    POWERS.  389 

2.  What  must  be  the  diameter  of  an  axle,  that  a  power  of 
1001b.,  applied  at  the  circumference  of  a  wheel  of  6  feet 
diameter,  may  balance  400  lb.  ? 

Inclined  Plane. 

444.  The  inclined  plane  is  nothing  more  than  a  slope  or 
l(.'clivity,  which  is  used  for  the  purpose  of  raising  weiglits.  It 
is  not  difficult  to  see  that  a  weight  can  be  forced  up  an  in- 
clined plane,  more  easily  than  it  can  be  raised  in  a  vertical 
line.  But  in  this,  as  in  the  other  machines,  the  advantage  is 
obtained  by  a  partial  loss  of  power. 

Thus,  if  a  weight  W, 
be  supported  on  the  in- 
clined plane  ABC,  by  a 
cord  passing  over  a  pulley 
at  F,  and  the  cord  from 
the  pulley  to  the  weight  be  parallel  to  the  length  of  the 
plane  AB,  the  power  P  will  balance  the  weight  W.,  when 

P    :   W    :  :   height  BC    :   length  AB. 

It  is  evident  that  the  power  ought  to  be  less  than  the 
weight,  since  a  part  of  the  weight  is  supported  by  the  plane  : 
hence, 

Rule. —  The  power  is  to  the  weight ,  as  the  height  of  the 
plane  is  to  its  length. 

Examples. 

1.  The  length  of  a  plane  is  30  feet, 'and  its  height  6  feet 
what  power  will  be  necessary  to  balance  a  weight  of  2001b.? 

2.  The  height  of  a  plane  is  10  feet,  and  the  length  20  feet 
what  weight  will  a  power  of  501b.  support? 

3.  The  height  of  a  plane  is  15  feet,  and  length  45  feet: 
what  power  will  sustain  a  weight  of  1801b.? 


390  MECHANICAL   POWERS. 

The  Wedge. 

445.  The  wedge  is  composed  of  two  inclined  planes,  united 
together  along  their  bases,  and  form- 
ing a  solid  ACB.  It  is  used  to 
cleaye  masses  of  wood  or  stone.  The 
resistance  which  it  overcomes  is  the 
attraction  of  cohesion  of  the  body 
which  it  is  employed  to  separate. 
The  wedge  acts,  principally,  by  being 
struck  with  a  hammer,   or  mallet,  on 

its    head,  and  very  Uttle    effect   can   be    produced   with   it,  by 
mere  pressure. 

All  cutting  instruments  are  constructed  on  the  principle  of 
the  inclined  plane  or  wedge.  Such  as  have  but  one  sloping 
edge,  like  the  chisel,  may  be  referred  to  the  inclined  plane  ;  and 
such  as  have  two,  like  the  ax  and  the  knife,  to  the  wedge. 

Rule. — Half  the  thickness  of  the  head  of  the  wedge,  is  to 
the  length  of  one  of  its  sides,  as  the  power  which  acts  against 
its  head  to  the  effect  produced  at  its  side. 

Examples. 

1.  If  the  head  of  a  wedge  is  4  inches  thick,  and  the  length 
of  one  of  its  sides  12  inches,  what  will  measure  the  effect  of 
a  force  denoted  by  96  pounds  ? 

2.  If  the  head  of  a  wedge  is  6  inches  thick,  the  length  of 
the  side  2t  inches,  and  the  force  applied  measures  250  pounds, 

what  will  be  the  measure  of  the  effect? 

♦ 

3.  If  the  head  of  a  wedge  is  9  inches,  and  the  length  of 
the  side  2  feet,  what  will  be  the  effect  of  a  blow  denoted  by 
200  pounds? 

4.  If  the  head  of  a  wedge  is  10  inches,  and  the  length  of 
the  side  30  inches,  what  will  measure  the  effect  of  a  blow 
denoted  by  600  ? 


MECHANICAL    POWERS.  391        . 


The  Screw. 


446.  The  screw  is  composed  of  two  parts — the  screw,  S,  and 
the  nut,  N. 

The  screw,  S,  is  a  cylinder 
with  a  spiral  projection  winding 
around  it.  The  nut,  N,  is  per- 
forated to  admit  the  screw,  and 
within  it  is  a  grove  into  which 
the  thread  of  the  screw  fits 
closely. 

The  handle,  D,  which  projects 
from  the  nut,  is  a  lever  which 
works  the  nut    upon  the  screw. 

The  power  of  the  screw  depends  on  the  distance  between  the 
threads.  The  closer  the  threads  of  the  screw,  the  greater  will 
be  the  power  ;  but  then  the  number  of  revolutions  made  by 
the  handle,  D,  will  also  be  proportionably  increased :  so  that 
we  return  to  the  general  principle — what  is  gained  in  power  is 
lost  in  time.  The  power  of  the  screw  may  also  be  increased 
by  lengthening  the  lever,  D,  attached  to  the  nut. 

The  screw  is  used  for  compression,  and  to  raise  heavy 
weights.  It  is  used  in  cider  and  wine  presses,  in  coining,  and 
for  a  variety  of  other  purposes. 

Riile. — As  the  distance  between  the  threads  of  a  screw,  is 
to  the  circumference  of  the  circle  described  by  the  power,  so 
is  the  poioer  employed  to  the  weight  raised. 

Examples. 

1.  If  the  distance  between  the  threads  of  a  screw  is  half  an 
inch,  and  the  circumference  described  by  the  handle  15  feet, 
what  weight  can  be  raised  by  a  power  denoted  by  120  pounds  ? 

2.  If  the  threads  of  a  screw  are  one-third  of  an  inch  apart, 
and  the  handle  is  12  feet  long,  what  power  must  be  appUed 
to  sustain  2  tons? 


392  QUESTIONS   IN    PHILOSOPHY. 

3.  What  force  applied  to  the  handle  of  a  screw  10  feet 
long,  with  threads  one  inch  apart,  working  on  a  wedge  whose 
head  is  5  inches,  and  length  of  side  30  inches,  will  produce  an 
effect  measured  by  100001b.? 

4.  If  a  power  of  300  pounds  applied  at  the  end  of  a  lever 
15  feet  long  will  sustain  a  weight  of  282^44  lb.,  what  is  the 
listance  between  the  threads  of  the  screw  ? 


QUESTIONS  IN  NATUEAL  PHILOSOPHY. 
UNIFORM   MOTION. 

447.  If  a  moving  body  passes  over  equal  spaces  in  equal 
portions  of  time,  it  is  said  to  move  with  uniform  motion,  or 
uniformly. 

448.  The  velocity  of  a  moving  body  is  measured  by  the 
space  passed  over  in  a  second  of  time. 

449.  The  space  passed  over  in  any  time  is  equal  to  the  pro- 
duct of  the  velocity  multiplied  by  the  number  of  seconds  in 
the  time. 

If  we  denote  the  velocity  by  Y,  the  space  passed  over  by 
S,  and  the  time  by  T,  we  have 

S  =  Y  X  T. 

LAWS    OF   FALLING    BODIES. 

450.  A  body  falling  vertically  downward  in  a  vacuum,  falls 
through  IGj'jft.  during  the  first  second  after  leaving  its  place 
of  rest,  48ift.  during  the  second  second,  80^  ft.  the  third 
second,  and  so  on :  the  spaces  forming  an  arithmetical  pro- 
gression of  which  the  common  difference  is  321  ft.,  or  double  the 
space  fallen  through  during  the  first  second.  This  numljer  is 
called  the  measure  of  the  force  of  gravity,  and  is  denoted  by  g. 

451.  It  is  seen  from  the  above,  that  the  velocity  of  a  body 


QUESTIONS  IN   PHILOSOPHY  393 

is    continually    increasing.       If    H     denote    the    height    fallen 

through,    T,    the    time,  V,   the   velocity   acquired,    and    g,   the 

force  of  gravity,   the   following  formulas   have   been  found   to 

express  the  relations  between  these  quantities  : 

Y  =    g   XT     .     .     .     (1). 

Y'  =  2g   XU     .     .     .     (2). 

H  =  |Y  X  T     .     .     .     (3). 

R  =:^g   XT     .     .     .     (4). 

From  which  we  see, 

1st.  That  the  velocity  acquired  at  the  end  of  any  time,  is 
<  qual  to  the  force  of  gravity  (32  Jr)  multiplied  by  the  time. 

2d.  That  the  square  of  the  velocity  is  equal  to  twice  the 
force  of  gravity  multijjlied  by  the  height;  or,  the  velocity  is 
equal  to  the  square  root  of  that  quantity. 

od.  That  the  space  fallen  through  is  equal  to  one-half  the 
velocity  multiplied  by  the  time. 

4th,  That  the  sjmce  fallen  through  is  equal  to  one-half  the 
force  of  gravity  multiplied  by  the  square  of  the  time. 

452.  If  a  body  is  thrown  vertically  upward  in  a  vacuum, 
its  motion  will  be  continually  retarded  by  the  action  of  gravi- 
tation. It  will  finally  reach  the  highest  point  of  its  asceht, 
and  then  begin  to  descend.  The  height  to  which  it  will  rise 
may  be  found  by  the  second  formula  in  the  preceding  para- 
graph, when  the  velocity  with  which  it  is  projected  upward  is 
known  ;   for  the  times  of  ascent  and  descent  will  be  equal. 

453.  The  above  laws  are  only  approximately  true  for  bodies 
falling  through  the  air,  in  consequence  of  its  resistance.  We 
may  measure  the  depths  of  wells  or  mines,  and  the  heights  of 
elevated  objects  approximately,  by  using  dense  bodies,  as  leaden 
bullets  or  stones,  which  present  small  surfaces  to  the  air. 

Examples. 
1.    A  body  has  been  falling  12  seconds  :    what   space  did  it 
describe  m  the  last  secoid,  and  what  in  the  whole  time? 

17* 


39i  QUESTIONS    IN    PHILOSOPHY. 

2.  A  body  has  been  falling  15  seconds  :  find  the  space 
described  and  the  velocity  acquired. 

3.  How  far  must  a  body  fall  to  acquire  a  velocity  of  120 
feet? 

4.  How  many  seconds  will  it  take  a  body  to  fall  throuo^h  a 
space  of  100  feet? 

5.  Find  the  space  through  which  a  heavy  body  falls  in  lO 
seconds,  and  the  velocity  acquired. 

6.  How  far  must  a  body  fall  to  acquire  a  velocity  of  1000 
feet? 

7.  A  stone  is  dropped  into  a  well,  and  strikes  the  water  in 
3.2  seconds  :    what  is  the  depth  of  the  well  ? 

8.  A  stone  is  dropped  from  the  top  of  a  bridge,  and  strikes 
the  water  in  2.5  seconds  :    what  is  the  height  of  the  bridge  ? 

9.  A  body  is  thrown  vertically  upward  with  a  velocity  of 
160  feet :  what  height  will  it  reach,  and  what  will  be  the 
time  of  ascent? 

10.  An  arrow  shot  perpendicularly  upward,  returned  again 
m  10  seconds.  Required  the  velocitv  with  which  it  was  shot, 
and  the  height  to  which  it  rose. 

11.  A  ball  is  let  fall  from  the  top  of  a  steeple,  and  reaches 
the  ground  in  three  seconds  and  a  half:  what  is  the  height  of 
the  steeple  ? 

12.  What  time  will  be  necessary  for  a  body  falling  freely, 
to  acquire   a  velocity  of  2500  feet  per  second? 

13.  If  a  ball  be  thrown  vertically  upward  with  a  velocity  of 
35v)  feet  per  second,  how  far  will  it  ascend,  and  what  will  be 
the  time  of  ascent  and  descent? 

14.  How  long  must  a  body  fall  freely  to  acquire  a  velocity 
of  3040  feet  per  second  ? 

15.  If  a  body  falls  freely  in  a  vacuum,  what  will  be  its 
velocity  afte*  45  seconds  of  fall? 


QUESTIONS    IN    rJIILOSOPIIY.  304 

16.  During  how  many  seconds  must  a  body  fall  in  a  vacuum 
to  acquire  a  velocity  of  1970  feet,  which  is  that  of  a  camion- 
ball? 

n.  What  time  is  required  for  a  body  to  fall  in  a  vacuum, 
from  an  elevation  of  3280  feet? 

18.  From  what  height  must  a  body  fall  to  acquire  a  vo- 
ocity  of  984  feet  ? 

19.  A  rocket  is  projected  vertically  upward  with  a  velocity 
of  380  feet :  after  what  time  will  it  begin  to  fall,  and  to 
what  height  will  it  rise  ? 

SPECIFIC     GRAVITY. 

454.  The  specific  gravity  of  a  body  is  the  weight  of  a 
unit  of  volume  compared  with  the  weight  of  a  unit  of  the 
tandard.  Distilled  rain-water  is  the  standard  for  measuring  the 
i>ccific  gravity  of  bodies.  Thus,  1  cubic  foot  of  distilled  rain- 
water weighs  1000  ounces  avoirdupois.  If  a  piece  of  stone,  of 
Uic  same  volume,  weighs  2500  ounces,  its  specific  gravity  is 
2.5  ;  that  is,  the  stone  is  2.5  times  as  heavy  as  water. 

If,  then,  we  denote  the  standard  by  1,  the  specific  gravity 
of  all  other  bodies  will  be  expressed  in  terms  of  this  standard; 
and  if  we  multiply  the  number  denoting  the  specific  gravity  of 
any  body  by  1000,  the  product  will  be  the  weight  in  ounces 
of  1  cubic  foot  of  that  body. 

If  any  body  be  weighed  in  air  and  then  in  water,  it  will 
weigh  less  in  water  than  in  air.  The  difference  of  the  weights 
will  be  equal  to  the  sustaining  force  of  the  water,  which  is 
found  to  be  equal  to  the  weight  of  an  equal  volume  of  water: 
hence. 

Rule. — If  we  know  or  can  find  the  weight  of  a  body  in 
air  and  in  water,  the  difference  of  these  lueights  will  he  equal 
to  that  of  an  equal  volume  of  water;  and  the  weight  of  the 
body  in  air  divided  by  this  difference  will  be  the  measure  of 
the  specific  gravity  of  the  body,  compared  with  woier  as  a 
standard. 


.  396 


QUESTIONS   IN   PHILOSOPHY. 

Table 

OF   SPECIFIC   GRAVITIES— WATER,    1. 


NAMES  OF  BODIES. 

SPEC.   GRAV. 

NAMES   OF  BODIES. 

SPKG   GBAV. 

METALS. 

21.000 

19.500 

13.500 

11.350 

10.51 

8.800 

8.758 

8.000 

7.800 

7.500 

7.291 

7.215 

3.10 
3.10 
3.00 
3.00 
2.93 
2.90 
2.75 
2.72 
2.69 
2.66 
2.62 
2.60 

PorDlivrv      

2.60 
2.50 
1.86 

1.049 

0.9859 

0.9822 

0.9476 

0.9452 

0.9206 

0.9121 

0.9036 

0.9036 

0.9012 

0.8941 

0.8614 

1.480 
2.050 
2.150 
2.542 
0.926 
1.842 
0.942 
0.969 

Platinum 

Brick 

Quicksilver 

WOODS. 

Oak,  fresh  felled 

White  Willow 

Box 

Lead 

Silver 

Bronze 

Elm 

Steel  

Horbeam 

Larch 

Tin        

Pine 

Zinc 

Maple 

BUILDING    STONES. 
Hornblende 

Ash 

Birch 

Fir 

Horse-chestnut 

SOLID    BODIES. 

.  Common  earth 

Moist  sand 

Basalt 

Alabaster 

Syenite 

Dolerite 

Gneiss 

Quartz 

Clay 

Limestone 

Flint 

Phonolite 

Ice 

Granite 

Lime 

Stone  for  building  . . . 
Trachyte 

Tallow 

Wax 

By  inspecting  this  Table,  we  see  the  weight  of  each  body 
compared  with  an  equal  volume  of  water.  Thus,  platinum  is  21 
times  as  heavy  as  water  ;  gold,  19  times  as  heavy  ;  iron,  t^ 
times  as  heavy,  &c. 

Examples  illustrating  Specific  G-ravity. 

1.  A  piece  of  copper  weighs  93  grains  in  air,  and  82 J  grains 
in  water  :  what  is  its  specific  gravity  ? 

2.  How  many  cubic  feet  are  there  in  2240  pounds  of  dry 
oak,  of  which  the  specific  gravity  is  .925,  a  cubic  foot  of 
standard  water  weighing  1000  ounces  ? 


QUESTIONS  IN    PHILOSOPHY.  397 

3.  A  piece  of  pumice-stone  weighs  in  air  50  ounces,  and 
when  it  is  connected  with  a  piece  of  copper  which  weighs  390 
ounces  in  air,  and  345  ounces  in  water,  the  compound  weighs 
344  ounces  in  water ;  what  is  the  specific  gravity  of  the 
stone  ? 

4.  A  right  prism  of  ice,  the  length  of  whose  base  is  20.45 
yards,  breadth  15.75  yards,  and  height  10.5  yards,  floats  on 
the  sea;  the  specific  gravity  of  the  ice  is  .930,  and  that  of  the 
sea-water  1.026  :  what  is  the  height  of  the  prism  above  the 
surface  of  the  water? 

5.  A  vessel  in  a  dock  was  found  to  displace  6043  cubic 
feet  of  water :  what  was  the  weight  of  the  vessel,  each  cubic 
foot  of  the  water  weighing  03  pounds  ? 

0.  A  piece  of  glass  was  found  to  weigh  in  the  air  33 
ounces,  and  in  the  water  21  ounces  :  what  was  its  specifiti 
gravity  ? 

7.  A  piece  of  zinc  weighed  in  the  air  IT  pounds,  and  lost 
when  weighed  in  water  2.35  pounds  :  what  was  its  specific 
gravity  ? 

8.  If  a  piece  of  glass  weighed  in  water  loses  318  ounces  of 
its  weight,  and  weighed  in  alcohol  loses  250  ounces,  what  is 
tlie  specific  gravity  of  the  alcohol? 

9.  A  flask  filled  with  distilled  water  weighed  14  ounces ; 
filled  with  brandy,  it  weighed  13.25  ounces  ;  the  flask  itself 
weighed  8  ounces  :  what  was  the  specific  gravity  of  the 
brandy  ? 

10.  What  is  the  weight  of  a  cubic  foot  of  statuary  marble, 
of  which  the  specific  gravity  is  2.83T,  the  cubic  foot  of  water 
weighing  1000  ounces  ? 

11.  A  jar  containing  air  weighed  24  ounces  33  grains  ;  the 
air  was  then  excluded,  and  the  jar  weighed  24  ounces ;  the 
jar  being  then  filled  with  oxygen  gas,  weighed  24  ounces  36.4 
grains  :  what  was  the  specific  gravity  of  the  oxygen  the  air 
being  taken  as  the  standard? 


g98  QUESTIONS   IN    PHILOSOPHY. 


mariotte's   law. 


455.  This  law,  which  relates  to  air  and  all  other  gases, 
steam  i  ad  all  other  vapors,  was  discovered  by  the  Abbe 
Mariottc,  a  French  philosopher,  who  died  in  1684.  It  will  be 
easily  understood  from  a  particular  example. 

Suppose  an  upright  cylindrical  vessel  in  a  vacuum  contains 
gas  which  is  confined  in  the  vessel  by  a  piston  at  the  upper. 
r,nd.  Suppose  the  gas  or  vapor  fills  the  whole  vessel,  and  the 
|)lston  is  loaded  with  a  weight  of  5  pounds.  If  now,  the 
piston  be  loaded  with  a  weight  of  10  pounds,  the  gas  will  be 
compressed  and  occupy  only  half  its  former  space.  If  the 
weight  be  increased  to  15  pounds,  the  gas  will  have  only  one- 
third  of  its  original  volume,  and  so  on.  At  the  same  time, 
the  density  of  the  gas  or  vapor  will  be  doubled,  made  three 
times  as  great,  and  so  on.  The  law,  therefore,  may  be  thus 
stated : 

Rule. —  The  temperature  remaining  the  same,  the  volume 
of  a  gas  or  vapor  is  inversely  proportional  to  the  pressure 
which  it  sustains.  Also,  the  density  of  a  gas  or  vapor  is  di- 
rectly proportional  to  the  pressure. 

Examples. 

1.  A  vase  contains  4.3  quarts  of  air,  the  pressure  being  10 
pounds  :  what  will  be  the  volume  of  the  air  when  the  pressure 
is  12.3  pounds,  the  temperature  remaining  the  same? 

2.  Under  a  pressure  of  15  pounds  to  the  square  inch,  a 
certain  quantity  of  gas  occupies  a  volume  of  20  quarts  :  what 
pressure  must  be  applied  to  reduce  the  volume  to  8  quarts? 

3.  A  quart  of  air  weighs  2.6  grains  under  a  pressure  of  15 
pounds  :  what  will  be  the  weight  of  a  quart,  if  the  pressure 
be  reduced  to  14.2  pounds  ? 

4.  The  pressure  upon  the  steam  contained  in  a  cylinder  is 
increased  from  25  pounds  upon  the  square  inch  to  4t  pounds  : 
what  part  of  the  original  volume  will  be  occupied  ? 


APPENDIX. 


DIFFERENT    KINDS    OF    UNITS. 
I.    ABSTRACT   UNITS. 

456.  The  abstract  unit  I  is  the  base  of  all  numbers,  and  is 
tailed  a  unit  of  the  first  order.  The  unit  1  ten,  is  a  unit  of 
the  second  order  ;  the  unit  1  hundred,  is  a  unit  of  the  third 
order  ;  and  so  for  units  of  the  higher  orders.  These  are  ab- 
stract numbers  formed  from  the  unit  1,  according  to  the  scale 
of  tens.  All  abstract  integral  numbers  are  collections  of  the 
unit  one. 

II.    UNITS  OF  CURRENCY. 

457.  Ill  all  civilized  and  commercial  countries,  great  care  is 
taken  to  fix  a  standard  value  for  money,  which  standard  is 
culled  the   Unit  of  Currency. 

In  the  United  States,  the  unit  of  currency  is  1  dollar  ;  in 
Great  Britain  it  is  I  pound  sterling,  equal  to  $4.84  ;  in  France 
it  is  1  franc,  equal  to  I8j%  cents  nearly.  All  sums  of  money 
are  expressed  in  the  unit  of  currency,  or  in  units  derived 
from  the  unit  of  currency,  and  having  fixed  proportions  to  it. 

III.    UNITS  OF  LENGTH. 

458.  One  of  the  most  important  units  of  measure  is  that 
for  distances,  or  for  the  measurement  of  length.  A  practical 
want  has  ever  been  felt  of  some  fixed  and  invariable  stand- 
ard, with  which  all  distances  may  be  compared :  such  fixed 
standard  has  been  sought  for  in  nature. 

There  are  two  natural  standards,  either  of  which  affords  this 
desired  natural  element.  Upon  one  of  them,  the  English  have 
founded  their  system  of  measures,  from  which  ours  is  taken  ; 
and    upon    the    other,    the    French    have    based    their    system. 


400  APPENDIX. 

These  two  systems,  being  the  only  ones  of  importance,  will  be 
alone  considered. 

First. — The  English  system  of  measures,  to  which  ours  con- 
forms, is  based  upon  the  law  of  nature,  that  the  force  of 
gravity  is  constant  at  the  same  point  of  the  earth's  surface, 
and  consequently,  that  the  length  of  a  pendulum  which  oscil 
ates  a  certain  number  of  times  in  a  given  period,  is  also  con 
stant.  Had  this  unit  been  known  before  the  adoption  and  use 
of  a  system  of  measures,  it  would  have  formed  the  natural 
unit  for  divisior  and  been  the  natural  base  of  the  system  of 
linear  measure.  But  the  foot  and  inch  had  long  been  used  as 
units  of  linear  measure  ;  and  hence,  the  length  of  the  pendu- 
lum, the  new  and  invariable  standard,  was  expressed  in  terms 
of  the  known  units,  and  found  to  be  equal  to  39.1393  inches. 
The  new  unit  was  therefore  declared  invariable — to  contain 
39.1393  equal  parts,  each  of  which  was  called  an  inch;  12  of 
these  parts  were  declared  by  act  of  Parliament  to  be  a  stand- 
ard foot,  and  36  of  them  an  Imperial  yard.  The  Imperial 
yard  and  the  standard  foot  are  marked  upon  a  brass  bar,  at 
the  temperature  of  62^°,  and  these  are  the  linear  measures 
from  which  ours  are  taken.  The  comparison  has  been  made 
by  means  of  a  brass  scale  82  inches  long,  manufectured  by 
Troughton,  in  London,  and  now  in  the  possession  of  the 
Treasury  Department. 

Second. — The  French  system  of  measures  is  founded  upon 
the  principle  of  the  invariability  of  the  length  of  an  arc  of 
the  same  meridian  between  two  fixed  points.  By  a  very 
minute  survey  of  the  length  of  an  arc  of  the  meridian  from 
Dunkirk  to  Barcelona,  the  length  of  a  quadrant  of  the  me- 
ridian was  computed,  and  it  has  been  decreed  by  the  French 
aw  that  the  ten-millionth  part  of  this  length  shall  be  regarded 
as  a  standard  French  metre,  and  from  this,  by  multiplicatioa 
and  division,  the  entire  system  of  linear  measures  has  been 
established. 

On   comparing  the  two   scales  very  accurately,   it  has   been 


UNl'Ib  OJ^   VOLUME.  401 

found  that  the  French  m^tre  is  equal  to  39.31079  English 
inches — differing  somewhat  from  the  Englisli  yard.  This  rela- 
tion enables  us  to  convert  all  measures  in  either  system  into 
the  corresponding  measures  of  the  other. 

IV.  UNITS  OF  SURFACK 

459.  The  linear  unit  having  been  established,  the  most  con- 
venient UNIT  OF  SURFACE  is  the  area  of  a  square,  one  of  whose 
sides  is  the  unit  of  length.  Thus,  the  units  of  surface  in  com- 
mon use,  are — 

•     A  square  inch  =  a  square  on  I  inch. 

A  square  foot  =  a  square  on  1  foot. 

A  square  yard  =  a  square  on  I  yard. 

A  square  rod  =  a  square  on  1  rod. 

V.  UNITS    OF  VOLUME. 

460.  The  unit  of  volume,  for  the  measurement  of  solids,  is 
taken  equal  to  the  volume  of  a  cube  one  of  whose  edges  is 
equal  to  the  linear  unit.  The  units  of  volume  in  common  use 
are — 

A  cubic  inch  =  a  cube  whose  edge  is  1  inch  ; 
A  cubic  foot  =  a  cube  whose  edge  is  1  foot  =  1728  cubic  in. 
A  cubic  yard  =  a  cube  whose  edge  is  1  yard  =  27  cubic  feet. 
A  perch  of  stone  =  24j  cubic  feet ; 

or  a  block  of  stone  1  ro^d  long,  1  foot  thick,  and  IJft.  wide. 
The  standard  unit  of  volume  for  the  measurement  of  liquids 
is  the  wine  gallon,  which  contains  231  cubic  inches. 

The  standard  unit  of  dry  measure  is  the  Winchester  bushel, 
which  contams  2150.4  cubic  inches,  nearly. 

VI.  UNITS  OF  WEIGHT. 

461.  Having  fixed  an  invaiiable  unit  of  length,  we  passed 
easily  to  an  invariable  unit  of  surface,  and  then,  to  an  invari- 
able unit  of  volume.  We  wish  now  to  define  an  invariable 
unit  of  weight. 


402  APPENDIX. 

It  has  been  found  that  distilled  rain-water  is  the  most  in- 
variable substance  ;  hence,  this  has  been  adopted  as  the  standard. 

We  have  two  units  of  weight,  the  avoirdupois  pound,  and 
the  pound  troy. 

The  standard  avoirdupois  pound  is  the  weight  of  2 1. 701554 
cubic  inches  of  distilled  water. 

The  standard  Troy  pound  is  the  weight  of  22.194422  cubic 
inches.  This  standard  is  at  present  kept  in  the  United  States 
Mint  at  Philadelphia,  and  is  the  standard  unit  of  weight. 

VII.    UNITS  OF  TIME. 

462.  Time  can  only  be  measured  by  motion.  The  diurnal 
revolution  of  the  earth  affords  the  only  invariable  motion  ; 
hence,  the  time  in  which  it  revolves  once  on  its  axis,  is  the 
natural  unit,  and  is  called  a  day.  From  the  day,  by  multipli- 
cation, we  form  the  weeks,  months,  and  years  ;  and  by  division, 
the  hours,  minutes,  and  seconds. 

VIII.    UNITS  OF  CIRCULAR  OR  ANGULAR  MEASURE. 

463.  This  measure  is  used  for  the  measurement  of  angles, 
and  the  natural  unit  is  the  right  angle.  But  this  is  not  the 
most  convenient  unit.  The  unit  chiefly  used  is  the  360  part 
of  the  circumference  of  a  circle,  called  a  degree,  which  is  di- 
vided into  60  equal  parts,  called  minutes,  and  these  again  into 
60  equal  parts,  called  seco7ids. 

Hemarks. 

464.  It  is  seen  that  all  the  units,  determined  by  the  pen- 
dulum, depend  on  time  as  the  ultimate  base  ;  that  is,  the 
length  of  a  pendulum  which  will  vibrate  seconds  determines  all 
^he  units  of  'tneasure  and  weight. 

Now,  time  is  measured  by  motion,  and  the  motion  of  the 
earth  on  its  axis  is  the  only  invariable  motion.  Hence,  we 
refer  to  this  to  fix  the  unit  of  time,  on  which  the  unit  of 
length  depends,  and  from  which  all  the  other  units  are  derived. 


ABSTRACT   UNITS.  4.03 

No  class  of  pupils  can  rightly  and  clearly  apprehend  the 
nature  of  numbers  and  the  operations  performed  upon  them, 
without  distinct  and  fixed  notions  of  the  units  ;  hence,  every 
teacher  should  labor  to  point  out  their  absolute  and  relative 
values  :    this  can  only  be  done  by  means  of  sensible  objects. 

Every    school-room,    therefore,    should    be   provided    with 
complete  set  of  all   the   denominate  units.     The  inch,  the  foot 
the   yard,  the  rod,  should    be   accurately  marked  off  on  a  con- 
spicuous part  of  the  room,  together  with  the  principal  units  of 
surface,  the  squre  inch,  square  foot,  square  yard,  &c. 

The  units  of  volume  should  also  be  exhibited.  The  cubic 
inch  and  the  cubic  foot  will  serve  as  illustrations  for  one  class 
of  the  units  of  volume  ;  and  the  pint,  quart,  gallon,  and 
bushel,  should  be  exhibited  to  illustrate  the  others. 

The  unit  of  weight  should  also  be  seen  and  handled.  A 
child  even  can  apprehend  what  is  meant  by  an  ounce  or  a 
pound,  when  it  takes  one  of  these  weights  in  its  hand  ;  and 
mature  years  can  acquire  the  idea  in  no  other  way. 

Let,  therefore,  every  school-room  be  furnished  with  a  com- 
plete set  of  models  to  illustrate  and  explain  the  absolute  and 
rdatiijc  values  of  the  different  units. 

I.    ABSTRACT  NUMBERS. 
465.   An  Abstract  Number  is  one  whose  unit  is  not  named. 

Table. 

10  Umts make  1  Ten. 

10  Tens 1  Hundred. 

10  Hundred 1  Thousand. 

10  Thousand 1  Ten-thousand. 

&c.,  &c. 

Table  Reversed. 

Ten.  TJnIta. 

nuncL  1  =  10. 

Thous.  1       =  10  =  100. 

Ton-thous.  1       =  10       =  100  =  1000. 

1     =     10     =     100     =     1000     =     10000. 
Scale. — Uniform — units,  10. 


4.01  APPENDIX. 

II.    CURRENCY. 

I.      UNITED    STATES   MOXEY. 

466.   United   States   Money   is   the  currency  established  by 
Congress,   a.  d.    1786.      The    names    or   denominations    of    its 
nits  are,  Eagles,  Dollars,  Dimes,  Cents,  and  Mills. 

The  coins  of  the   United   States  are  of  gold,  silver,  copper, 
nd  nickel,  and  are  of  the  following  denommations  : 

1.  Gold  :   Double-eagle,  eagle,   half-eagle,   three-dollars,  quar- 
ter-eagle, dollar. 

2.  Silver  :   Dollar,  half-dollar,  quarter-dollar,  dime,  half-dime, 
and  three-cent  piece. 

3.  Copper :    Cent. 

4.  Nickel:   Cent. 

*  Table. 

10  Mills    ....     make  1  Cent,   ....  marked  ct. 

10  Cents 1  Dime, d. 

10  Dimes 1  Dollar, |. 

10  Dollars 1  Eagle, B. 

20  Dollars 1  Double  Eagle.    .    .      2  E. 


Table  Reversed. 

ct 

m. 

d.         1 

= 

10. 

$ 

1   =     10 

= 

100. 

E. 

1   : 

=   10  =   100 

= 

1000. 

1 

=  10 

100  =   1000 

= 

10000. 

Scale. — Uniform — units,  10. 

In  all  the  States  the  shilling  is  reckoned  at  12  pence,  the 
^ound  at  20  shillings,  and  the  dollar  a^  100  cents. 

The  following  table  shows  the  number  of  shillings  in  a  dol 
lar,  the  value  of  iEl  in  dollars,  and  the  value  of  $1  in  the 
fraction  of  a  pound  : 


ENGLISH   MONEY.  4:05 

Til  English  currency,       4s.  Gd.  -  ^21  =  $4.84,    and  |1  re  £^\^. 


In  New  York,  Ohio, )     „  ^,       ^^i  ,  *, 

,,.  , .  '  ^    8s.        -  £1=  $21,       and  $1  =  M. 

Michigan,  )  ^  * 

North  Carolina,  10s. 

r„KJ„Pa..   DeI.,J    ,,«,..  ^i^^2^_      aud|l  =  ^. 

In  S.  CaroUna  &  Ga.,       4s.  8d.  -  £1  =  $4|,       and  $1  =  £^, 

In  Canada  and  Nova,)     ^  ^,       *.  ,  <,^,        ^, 

„    ,.  7    5s.        -  iei  =  $4,         and  $1  =  £i, 

Scotia,  3  * 

Notes. — The  present  standard  or  degree  of  purity  of  the  coins  was 
fixed  by  Act  of  Congress  in  1837.     It  is  this: 

1.  Nine  hundred  equal  parts  of  pure  gold,  are  mixed  with  100 
parts  of  alloy,  of  copper,  and  silver,  (of  wliich  not  more  than  one-half 
must  be  silver)  thus  forming  1000  parts,  equal  to  each  other  in 
weight.  The  silver  coins  contain  900  parts  of  pure  silver,  and  100 
parts  of  pure  copper.  The  copper  coins  are  of  pure  copper.  The 
nickel  cent  is  88  parts  copper  and  12  nickel. 

2.  The  eagle  contains  258  grains  of  standard  gold,  and  the  other 
gold  coins  in  the  same  proportion.  The  dollar  contains  412i  grains 
of  standard  silver,  and  the  others  in  the  same  proportion.  The  cent, 
168  grains  of  pure  copper. 

3.  If  a  given  quantity  of  gold  or  silver  be  divided  into  24  equal 
parts,  each  part  is  called  a  carat.  If  any  number  of  carats  be  mixed 
with  so  many  equal  carats  of  a  less  valuable  metal,  that  there  be  24 
carats  in  the  lAixture,  then  the  comx)ound  is  said  to  be  as  many 
carats  fine  as  it  contains  carats  of  the  more  precious  metal,  and  to 
contain  as  much  aUoy  as  it  contains  carats  of  the  baser. 

4.  Although  the  currency  of  the  United  States  is  in  dollars,  cents, 
and  mills,  yet  in  some  of  the  States  the  old  currency  of  pounds 
shillings,  and  pence,  is  still  nominally  preserved. 

II.     ENGLISH    MONEY. 

467.  The  units  or  denominations  of  English  money  are 
ejuineas,  ponnds,  shillings,  pence,  and  farthings. 


•.too  ArricNDix. 

Table. 

4  fartliings,  marked  far.,  make  1  penny,       marked       d. 

12  pence, 1  sliilling,  "  ». 

20  shillings, 1  pound,  or  sovereign,  £. 

21  shillings, 1  guinea. 

Table  Reversed. 

d.  fax, 

1     =  4. 

£               1     =       12     =  48. 

1     =     20     =     240     =  960. 

Notes. — 1.  The  primary  unit  in  English  money  is  1  farthing. 
The  units  of  the  scale,  in  passing  from  farthings  to  pence,  are  4 ;  in 
passing  from  pence  to  shillings,  the  units  of  the  scale  are  12 ;  in  pass- 
ing from  shillings  to  pounds,  they  are  20. 

2,  Farthings  are  generally  expressed  in  fractions  of  a  penny.  Thus, 
Ifar.  =  id. ;  2  far.  =  id. ;  3  far.  =  |d. 

3.  The  standard  of  the  gold  coin  is  22  parts  of  pure  gold  and  2 
parts  of  copper. 

The  standard  of  silver  coin  is  37  parts  of  pure  silver,  and  3  parts 
of  copper. 

A  poimd  of  gold  is  worth  14.2878  times  as  much  as  a  pound  of 
silver.    In  copper  coin,  24  pence  make  1  pound  avoirdupois. 

By  reading  the  second  table  from  left  to  right,  we  can  see  the 
value  of  any  unit  expressed  in  each  of  the  lower  denominations. 
Thus,  Id.  =  4  far. ;  Is.  =  12d.  =  48  far.    £1  =  20s.  =  240d.  =  960  far. 

TABLE    OF    LEGAL   VALUES    OF   FOREIGN    COINS. 


Franc  of  France  and  Belgium.. 

Florin  of  the  Netherlands 

Guilder  of  do 

Livre  Tournois  of  France 

Milrea  of  Portugal 

Mih-ea  of  Madeira , 

Milrea  of  the  Azores 

Marc  Banco  of  Hamburg , 

Pound  Sterling  of  Great  liritain. 

Pagoda  of  India , 

Real  Vellon  of  Spain 

Real  Plate  of       do 

Rupee  Company 


$ 

ct. 

0 

18^ 

40 

40 

18-i 

1 

12 

1 

00 

831 

35 

4 

84 

1 

84 

05 

10 

44| 

VALLK    OF    COINS. 


407 


1^111)60  of  Britisli  Jiulia 

Jiix  Dollar  of  Denmark 

Jiix  Dollar  of  Prussia 

Kix  Dollar  of  Bremen 

Kouble,  silver,  of  Russia 

Tale  of  China 

Dollar  of  Sweden  and  Norway 

Si>ecie  Dollar  of  Denmark 

Dollar  of  Prussia  and  Northern  States  of  Germany... 

Florin  of  Southern  States  of  Germany 

Florin  of  Austria  and  City  of  Augsburg 

Lira  of  the  Lombardo-Venetian  Kingdom 

Lira  of  Tuscany 

Lira  of  Sardinia 

Ducat  of  Naples 

Ounce  of  Sicily 

Pound   of  Nova   Scotia,   New  Brunswick,  Newfound 
land,  and  Canada 


44i 

1 

00 

i)8.\ 

T8| 

75 

1 

48 

1 

OG 

1 

05 

69 

40 

. 

481 

16 

16 

18  ,-•'3 

80 

2 

40 

4 

04 

TABLE    OF   FOREIGN   COINS    OF   USAGE    VALUES. 


Berlin  Rix  Dollar 

Current  marc 

Crown  of  Tuscany  . . . 
Elberfeldt  Rix  Dollar 

Florin  of  Saxony 

"         Bohemia 

Elberfddt. . . 

"        Prussia 

Trieste 

"         Nuremburg. 
Frankfort.., 

"         Basil 

St.  Gaul . . . . 

Creveld 

Livre  

do 

do 


l^lorence 

Genoa 

Geneva 

Jamaica  Pound 

Leghorn  Dollar 

Leghorn  Livre  (GJ  to  the  doUar) 

Livre  of  Catalonia 

Neulchatel  Livre 

Pezza  of  Leghorn 

Rhenish  Rix  Dollar 

Swiss  Li\Te 

Scuda  of  Malta 

Turkish  Piastre 


ct 

69i 

28 

05 

69| 

48 

48 

40 

221 

48 

40 

40 

41 

40 
15 

18| 

21 

00 

90 

1511 

53i 

26i 

90 

60} 

27 

40 

05 


[Tlie  above  Tables  arc  taken  from  a  work  en  tho  Tariff,  by  E.  D.  Ogden,  Esq.,  of 
Misf  ^e\v  York  Custom-house]. 


408  APPENDIX. 

III.    LINEAR  MEASURE. 

I.     LONG     MEASURE. 

468.   This   measure    is    used    to   measure  distances,   lengths, 
breadths,  heights,  and  depths. 

Table. 

12  inches make    1  foot, marked  ft. 

3  feet 1  yard, yd. 

51  yards,  or  IGi  feet      ....    1  rod,  rd. 

40  rods 1  furlong, fur. 

8  furlongs,  or  320  rods    ...    1  mile, mi. 

3  miles 1  league, L. 

69^  statute  miles,  nearly,  or    .      )  1  degree  on  the  equator,; 

60  geographical  miles      ...      )  or  any  great  circle, 

360  degrees a  circumference  of  the  earth. 


Table   Reversed. 


[  deg., 


a 

in. 

' 

yd. 

1 

12. 

rd. 

1   = 

3 

36. 

far. 

1 

= 

H  = 

161 

198. 

mi. 

1   = 

40 

= 

220  = 

660 

1920. 

1   = 

=  8  = 

320 

= 

neo    = 

5280 

63360. 

Notes. — 1.  "A  fathom  is  a  length  of  six  feet,  and  is  generally  used 
to  measure  the  depth  of  water. 

2.  A  hand  is  4  inches,  and  is  used  to  measure  the  height  of 
horses ;  a  common  pace  is  3  Teet ;  a  military  pace,  2}  feet ;  a  geo- 
graphical mile  equals  a  minute  of  a  great  circle;  a  knot  (used  by 
sailors)  is  a  geographical  mile. 

3.  The  units  of  the  scale,  in  passing  from  inches  to  feet,  are  12, 
in  passing  from  feet  to  yards,  3;  from  yards  to  rods,  5i;  from  rods 
to  furlongs,  40;  and  from  furlongs  to  miles,  8. 

ENGLISH    SYSTEM. 

469.  The  Imperial  yard  of  Great  Britain  is  the  one  from 
which  ours  is  taken.     Hence,  the  units  of  measure  are  identical. 


CLOTH   MEASURE.  409 

FRENCH    SYSTEM. 

470.  The  base  of  the  new  French  system  of  measures  is  the 
measure  of  the  meridian  of  the  earth,  a  quadrant  of  which  is 
10,000,000  mitres,  measured  at  the  temperature  of  32°  Fahr. 
The  multiples  and  divisions  of  it  are  decimals,  viz. :  1  metre 
=  10  decunetres  =  100  centimetres  =  1000  millhnetres  — 
3.280899  United  States  feet,  or  39.3t0t9  inches. 

This  relation  enables  us  to  convert  all  measures  in  either 
system  into  the  corresponding  measures  of  the  other. 

Austrian,       1  foot  =  12.448  U.  S.  inches  =  1.03t3T  foot. 

Prussian,    }  ^  ^^^^  ^  j^.sci     "  "      =  1.0800       " 

Rhineland, ) 

Swedish,        1  foot  =11.690     "  "      =  0.9t4145  " 

f  1  foot  =  11.034     "  "      =  0.9195       " 

Spanish,      -J  league   (royal)  =  25000  Span.   ft.  =  4  J   miles  )  ^ 

(     "  (common)  =  19800  "        =  3i       "    )  I 

II.     CLOTH   MEASURE. 

471.  Cloth  measure  is  used  for  measuring  all  kinds  of  cloth, 
ribbons,  and  other  things  sold  by  the  yard. 

Table. 
2k  inches,  (in.)    .    .      make    1  nail,     ....    marked    na. 
4    nails 1  quarter  of  a  yard,  .    .    .    qr. 

3  quarters 1  Ell  Flemish,  .    .    .    .  E.  Fl. 

4  quarters 1  yard, yd. 

5  quarters 1  EU  EngUsh, E.R 

Table  Reversed. 

□a.  in. 

qr.  1       =  2i. 

KFL  1=4=9. 

ya.  1     =     3     =     12     =     2t. 

E.E.  1     =     li  =     4     =     16     =     36. 

1     =     IJ  =     If  =     5     =     20     =     45. 

Note. — The  units  of  the  scale,  in  this  measure,  are  2i,  4,  3,  \ 
and  f. 

18 


410 


APPENDIX. 


IV.    UNITS  OF  SUEFACE. 

I.     SQUARE    MEASURE. 

472.   Square  measure  is  used  in  measuring  land,  or  any  thiKg 
in  which  length  and  breadth  are  both  considered. 


Table. 

i  44  square  inches  {sq.  in.)  make  1  square  foot,  .... 

9  square  feet 1  square  yard,      .    .    . 

80i  square  yards 1  square  rod,  or  1  perch, 

40  square  rods,  or  40  perches,  1  rood, 

4  roods,  or  160  perches,      .    .  1  acre,     ... 

640  acres 1  square  mile 


JSq.ft 
Sq.yd. 

P. 

B. 

A. 

M. 


A.  1 

1     =     4 


p. 
1 

40 
160 


Table  Reversed. 

6q.  ft  sq.  in. 

1  =  144. 

9  =  1296. 

2121  =  39204. 

1210     =     10890  =  1568160. 

4840     =     43560  =  6272640. 


Bq.  yd. 
1       = 

30J  = 


Note. — The  units  of  the  scale  in  this  measure  are  144,  9,  80}, 
40,  and  4. 

II.    surveyors'  measure. 

473.  The  Surveyor's,  or  Gunter's  chain,  is  generally  used  in 
surveying  land.  It  is  4  poles,  or  66  feet  in  length,  and  is 
divided  into  100  links. 

Table. 

7^0%  inches    ....    make    1  link, marked    I. 

4  rods  =  66  ft.  =  100  links        .    1  chata, c. 

80  chains 1  mile, mi 

1  square  chain 16  square  rods,  or  perches,      .     .  P 

10  square  chains 1  acre, A 

Notes. — 1.  Land  is  generally  estimated  in  square  miles,  acres 
roods,  and  square  rods,  or  perches. 

2.  The  units  of  the  scale,  in  this  measure,  are  Ty'/o,  4,  80,  1, 
and  10. 


UNITS  OF  VOLUME.  411 

V.    UNITS  OF  VOLUJME,  OR  CAPACITY. 

I.     CUBIC   MEASURE. 

474.  Cubic  measure  is  used  for  measuring  stone,  timber, 
earth,  and  sucli  other  things  as  have  the  three  dimensions, 
cngtli,  breadth,  and  thickness. 

Table. 


1728  cubic  inches  {cu.  in.)  make  1  cubic  foot,           .    marked    cu.  ft. 

27  cubic  feet 1  cubic  yard, cu.  yd. 

40  feet  of  roimd,  or          ;  ^  ^                                                      _, 

^„  ,         ,  -        '    .    ,       h      .1  ton, T. 

50  feet  of  liewn  timber,) 

42  solid  feet 1  ton  of  shipping,      ....         T. 

8  cord  feet,   or)  1        /i                                                  n 

128  cubic  feet,      f       •    •    •    •        ^'  ' 

24i  cubic  feet  of  stone    .    ,    .    1  perch, P. 


Notes. — 1.  A  cord  of  wood  is  a  pile  4  feet  wide,  4  feet  bigh,  and 
8  feet  long. 

2.  A  cord  foot  is  1  foot  in  length  of  the  pile  which  makes  a  cord. 

3.  A  CUBE  is  a  solid  or  volxmie  bounded   by  six  equal  squares, 
called  faces;  the  sides  of  the  squares  are  called  edges. 

4.  A  cubic  foot  is  a  cube,  each  of  whose  faces  is  a  square  foot; 
its  edges  are  each  1  foot. 

5.  A  cubic  yard  is  a  cube,  each  of  whose  edges  is  1  yard 

6.  A  ton  of  round   timber,  when  square,  is  supposed  to  produce 
40  cubic  feet:  hence,  one-fifth  is  lost  by  squoHng. 


II.      LIQUID    MEASURE. 

475.  Liquid,   or   wine   measure,   is  used  for    measuring    all 
liquids  except  ale,  beer,  and  milk. 


4:12  APPENDIX. 


Table. 


4  giiiS  (gi.)    ....     make  1  pint, marked    pt 

2  pints 1  quart, qi 

4  quarts .1  gallon, gal. 

31i  gallons 1  barrel, bar.  or  bbl. 

42  gallons 1  tierce, tie?' 

63  gallons 1  hogsliead, Jihd 

2  hogsheads 1  pipe, ^)^ 

3  pipes,  or  4  hogsheads,    .  1  tun, tun. 


Table  Reversed. 

pt               gi. 

qt 

1   =         4. 

gal 

1    = 

2  =         8. 

bar.             1    = 

4    = 

8  =       32. 

tier. 

1  =  311  = 

126  = 

252  =  1008. 

hbd.         1 

=11       42  = 

168  = 

336  =   1344. 

pi. 

1    =    11 

=  2         63  = 

252  = 

504  =  2016. 

tun. 

1 

=   2  =   3 

=4       126  = 

504  = 

1008  =  4032. 

1    = 

2 

=  4   =   6 

=8       252  = 

1008  = 

2016  =  8064. 

Notes. — 1.  The  standard  unit,  or  gallon  of  wine  measure,  in  the 
United  States,  contains  t31  cubic  inches,  and  hence,  is  equal  to  the 
weight,  avoirdupois,  of  8.839  «ubie  '4nche8  of  distilled  water,  very 
nearly.  _       * 

2.  The  English  Imperial  wine  gallon  contains  277.274  cubic  inches, 
and  hence,  is  equal  to  1.2  times  the  wine  gallon  of  the  United 
States,  nearly. 

III.     BEER    MEASURE. 

476.  Beer  measi^e  was  formerly  used  for  measuring  ale, 
beer,  and  milk.  They  are  now  generally  measured  by  wine 
measure. 

Table. 

2  pints  {pt.)         .    .    .     make  1  quart, marked    gf. 

4  quarts 1  gallon, gal 

36  gallons 1  barrel, bar 

54  gallons -  1  hogshead, 7i7id 


DRY  MEASURE.  413 


Table  Reversed. 

gal. 
bar.                  1       = 

qt 
1 
4 

= 

pL 

2. 

8. 

hhd. 

1       =       3G       = 

144 

= 

288. 

1   = 

IJ  =     54     = 

216 

= 

432. 

Notes.— 1.  The  standard  gallon,  beer  measure,  contains  282  cable 
mches,  and  hence,  is  equal  to  the  weight  of  10.1799  cubic  inches  o» 
distilled  rain-water. 

2.  Milk  is  generally  bought  and  sold  by  wine  measure. 


III.      DRY    MEASURE. 

477.   Dry  measure  is  used  iu  measuring  all  dry  articles,  such 
as  grain,  fruit,  salt,  coal,  &c. 

Table. 

2  pints  (pL)     ....     make  1  quart, marked    qt. 

8  quarts 1  peck, pk. 

4  pecks 1  bushel, Im, 

36  bushels 1  chaldron, cTi, 

Table  Reversed. 


qt 

pt 

pk. 

1 

= 

2. 

bn. 

1 

= 

8 

= 

16. 

Ob. 

1 

= 

4 

= 

32 

= 

64. 

1      = 

=     36 

= 

144 

= 

1152 

= 

2304. 

Notes. — 1.  The  standard  hushd  of  the  United  States  is  the  Win 
diester  bushel  of  England.  It  is  a  circular  measure  18i  inches  it 
diameter,  and  8  inches  deep,  and  contains  2150.4  cubic  inches,  nearly 
It  contains  77.627413  pounds  avoirdupois  of  distilled  water. 

2.  A  gallon,  dry  measure,  contains  268.8  cubic  inches. 

3.  Wine  measure,  beer  measure,  and  dry  measure,  and  all  meas- 
ures of  volume,  differ  from  the  cubic  measure  only  in  the  unit  which 
is  used  as  a  standard. 


4:14  APPENDIX. 

VI.    UNITS  OF  WEIGHT. 

I.     AVOIRDUPOIS    WEIGHT. 

478.  By  this  weight  all  coarse  articles  are  weighed,  such  as 
hay,  grain,  chandlers'  wares,  and  all  metals  except  gold  and 
silver. 

Table. 

16  drams  {di\)  ....      make  1  ounce,         ....      marked    oz. 

16  ounces 1  pound, lb. 

25  pounds 1  quarter, qr- 

4  quarters 1  hundred  weight cwt. 

20  hundred  weight 1  ton, T. 

Table  Reversed. 


ot 

dr. 

lb. 

1 

= 

16. 

qr. 

1 

^zz 

16 

=z 

256. 

1   = 

25 

=: 

400 

= 

6400. 

4  = 

100 

= 

1600 

= 

25600. 

ewt 
T.  1      = 

1     =     20     =     80     =     2000     =     32000     =     512000. 

Notes. — ^1.  The  standard  avoirdupois  pound  is  the  weight  of 
27.7015  cubic  inches  of  distilled  water;  and  hence,  1  cubic  foot 
weighs  1000  ounces,  very  nearly. 

2.  By  the  old  method  of  weighing,  adopted  from  the  English  sys- 
tem, 112  pounds  were  reckoned  for  a  hundred  weight.  But  now,  the 
laws  of  most  of  the  States,  as  well  as  general  usage,  fix  the  hun- 
dred weight  at  100  pounds. 

3.  The  units  of  the  scale,  in  passing  from  drams  to  ounces,  are 
16;  from  ounces  to  pounds,  16;  from  pounds  to  quarters,  25;  from 
quarters  to  hundreds,  4;  and  from  hundreds  to  tons,  20. 

II.    TROY   WEIGHT. 

479.  Gold,  silver,  jewels,  and  liquors,  are  weighed  by  Troy 
velght. 

Table. 

•24  grains  ( 5'/'.)   .     .     .  make    1  pennyweight,      .     .    marked    ^wt. 

20  pennyweights 1  ounce, oz, 

12  ounces 1  pound,  ..•»••*«•     lb. 


apothecaries'  weight. 


Table 

Reversed. 

pwt. 

gf- 

oz. 

1         = 

24. 

lb. 

1 

z= 

20       = 

480. 

1 

=     12 

= 

240       = 

5T60. 

Notes.— 1.  The  standard  Troy  pound  is  tlio  weight  of  22.794377 
cubic  inches  of  distilled  water.  Hence,  it  is  less  than  the  pound 
avoirdupois. 

2.  7000    troy  grains  =      1  pound  avoirdupois. 

175    troy  pounds  =  144  pounds  " 

175    troy  ounces  =  192  ounces  " 

437i  troy  grains  =      1  ounce  " 

3.  The  Troy  pound  being  the  one  deposited  in  the  Mint  at  Phila- 
delphia, is  generally  regarded  as  the  standard  of  weight. 

4.  The  units  of  the  scale  are  24,  20,  and  12. 

III.    APOTHECARIES^    WEIGHT. 

480.  This  weight  is  used  by  apothecaries  and  physicians  iu 
mixing  their  medicines.  But  medicines  are  generally  sold,  in 
the  quantity,  by  avoirdupois  weight. 

Table. 

20  grains  (gr.) ....     make    1  scruple,       ....     marked  3. 

3  scruples 1  dram, 3. 

8  drams 1  ounce, f . 

12  ounces 1  pound, lb 

Table  Reversed. 


3 

gr. 

3 

1 

= 

20 

1 

3 

= 

60 

8       = 

24 

r= 

480 

ib  1       = 

1       =       12   .    =       96       =       288       =       5760 

Notes. — 1.  The  pound,  ounce,  and  grain,  are  the  same  as  the  pound, 
\mce,  and  grain,  in  Troy  weight. 

2.  The  units  of  the  scale,  in  passing  from  grains  to  scruples,  are 
20 ;  in  passing  from  scruples  to  drams,  3 ;  from  drams  to  ounces,  8 ; 
and  from  ounces  to  pounds,  12. 


416  APPENDIX. 

IV.    FRENCH    SYSTEM. 

481.  The  basis  of  tMs  system  of  weights  is  the  weight  in 
vacuo  of  a  cubic  decimetre  of  distilled  water.  This  weight  is 
called  a  kilogramme,  and  is  the  unit  of  the  French  system.  P 
is  equal  to  2.20413t  pounds  avoirdupois.  The  other  denom 
nations  are  as  follows : 

10  milligrammes  =  1  centigramme ;  10  centigronmes  =  1 
decigramme  ;  10  decigrammes  =  1  gramme  ;  10  grammes  =  1 
decagramme  ;  10  decagrammes  =  1  hectogramme ;  10  hecto- 
grammes =  1  kilogramme  ;  10  kilogrammes  =  1  quintal ;  10 
quintals  =  1  ton  of  sea-water. 

COMPARISON    OF    WEIGHTS. 


English^ 

1  pound 

=  1.000936 

pounds  avou'dupoia. 

French, 

1  kilogramme 

=  2.204131 

It                tt 

Spanish^ 

1  pound 

=   1.0152 

U                              it 

Swedish, 

1  pound 

=  0.9316 

tt                  tt 

Austrian, 

1  pound 

=  1.2351 

tt                     tt 

Prussian, 

1  pound 

=  1.0333 

tt                  It 

VII.    UNITS  OF  TIME. 
482.   Time  is  a   part   of  duration.    The  time   in  which  the 
earth  revolves  on  its  axis  is  called  a  day.     The  time  in  which 
it  goes  round  the  sun  is  called  a  solar  year.     Time  is  divided 
into  parts  accordmg  to  the  following 

Table. 

60  seconds,  see.                 make  1  minute,                      marked  m. 

60  minutes 1  hour, ,    ,  7ir. 

24  hours 1  day,    .......  da. 

7  days 1  week,     .    , wlc.^' 

52  weeks,  nearly 1  year, yr.^ 

12  calendar  months  =  365  da.  1  Julian  or  common  year, .    .  yr. 

366  days                               make  1  leap-year 

100  years 1  century,           cem. 


DATES. 


417 


Table  Reversed. 


wk. 

1        = 
521     = 


da. 

1  = 

1  = 

365  = 


hr.  1 

1   =  GO 

24   =  1440 

1G8   =  10080 

StGO   =  525600 


BCC 

60. 

3600. 

86400. 

604800. 

31536000. 


The  year  is  divided  into  12  calendar  months 


No.  No.  days. 

1st.   January,    .     .     .     .  31 

2d.    February,  ....  28 

Sd.    March, 31 

4th.  April, 30 

5th.  May, 31 

6th.  June, 30 


No. 

7th.  July,      .    . 

8th.  August, 

9  th.  September, 

10th.  October,     . 

11th.  November, 

12th.  December,  . 


No.  days. 

31 
31 
30 
31 
30 
31 


The  number  of  days  in  each  month  may  be  remembered  by 
the  following  : 

Thirty  days  hath  September, 
April,  June,  and  November  ; 
All  the  rest  have  thirty-one, 
Excepting  February,  twenty-eight  alone. 

Notes. — 1.  Days  are  numbered  in  each,  month  from  the  first  day 
of  the  month. 

2.  Months  are  numbered  from  January  to  December. 

3.  The  centuries  are  numbered  from  the  beginning  of  the  Cliristian 
Era.  The  year  30,  for  example,  at  its  commencement,  was  called  the 
30th  year  of  the  first  century,  though  neither  the  century  nor  the 
year  had  elapsed.  Thus,  Juno  2d,  ISoG,  was  the  6th  month  of  the 
5Gth  year  of  the  19th  century. 

4.  The  civil  day  begins  and  ends  at  12  o'clock  at  night.  Li  tlie 
civil  day,  the  hours  are  reckoned  from  that  time. 


Dates. 

1.  The  length  of  the  solar  year  is  365  da.  5  hr.  48  m.  48  sec,  very 
nearly.  It  is  desirable  to  have  the  periods  and  dates  of  the  civil  year 
corresiwnd  to  those  of  the  solar  year ;  else,  the  summer  months  of  tho 

18* 


4:18  APrENDlX. 

one  would  in  time  become  tlie  winter  months  of  the  other,  thereby 
producing  great  confusion  in  dates  and  history. 

2.  The  common  civil  year  is  reckoned  at  365  da.,  and  the  solar 
year  at  365  da.  6hr.  The  6  hours  accumulate  for  4  years  before  they 
are  counted,  when  they  amount  to  1  day,  and  are  added  to  February; 
and  the  year  is  called  a  bissextile,  or  Imp-year. 

3.  The  odd  6  hours  have  been  so  added,  that  the  leap-years  occur 
in  those  numbers,  which  are  divisible  by  4.  Thus,  1856,  1860,  1864 
&c.,  are  leap-years ;  and  when  any  number  is  not  divisible  by  4,  the 
remainder  denotes  how  many  years  have  passed  since  a  leap-year. 

4.  This  method  of  disposing  of  the  fractional  part  of  the  year 
would  be  without  error,  if  the  solar  year  were  exactly  365  da.  6  hr.  in 
length;  but  it  is  not;  it  is  only  365 da.  5hr.  48m.  48 sec.  long: 
hence,  the  leap-year  is  reckoned  at  too  much,  and  to  correct  this 
error,  every  centennial  year  is  reckoned  as  a  common  year.  But  this 
makes  an  error  again,  on  the  other  side,  and  every  fourth  centennial 
year  the  day  is  retained.  Thus,  1800  was  not,  and  1900  will  not  be, 
reckoned  a  leap-year:  the  error  will  then  be  on  the  other  side,  and" 
2000  will  be  a  leap-year.  This  disposition  of  the  fractional  part  of 
the  year  causes  the  civil  and  solar  years  to  correspond  very  nearly, 
and  indicates  the  following  rule  for  finding  the  leap-years: 

Rule. — Every  year  ivhich  is  divisible  by  4  is  a  leap  year, 
unless  it  is  a  centennial  year,  and  then  it  is  not  a  leap-year 
unless  the  number  of  the  century  is  also  divisible  by  4. 

5.  The  registration  of  the  days,  by  reckoning  the  civil  year  at 
365  da.,  was  established  by  the  Eoman  emperor,  Julius  Caesar,  and 
hence  this  period  is  sometimes  called  the  Julian  year. 

The  error,  arising  from  the  fractional  part,  continued  to  increase 
until  1582,  when  it  amounted  to  10  days;  that  is,  as  the  year  had 
been  reckoned  too  long,  the  number  of  days  had  been  too  few,  and  the 
count,  in  the  civil  year,  was  behind  the  count  in  the  solar  year. 

In  this  year  (1582),  Pope  Gregory  decreed  the  4th  day  of  October 
to  be  called  the  14th,  and  this  brought  the  civil  and  the  solar  years 
together.  The  new  calendar  is  sometimes  called  the  Gregorian 
Caleridar. 

6  The  method  of  dating  by  the  old  count,  is  called  Old  Style; 
and  by  the  new,  New  Style.  The  difference  is  now  12  days  In 
Russia,  they  still  use  the  old  style;  hence,  their  dates  are  12  days 
behind  ours.    Their  4th  of  January  is  our  16th. 


UNITS  OF   CIRCULAR   MEASURE. 


410 


VUI.    UNITS  OF  CmCULAK  MEASURE. 

483.    Angular,    or   circular    measure,    is   used  iu 
latitude    and    longitude,    iu    measuring    the    motions    of    the 
heavenly  bodies,  and  also  in  measuring  angles. 

The  circumference  of  every  circle  is  supposed  to  be  divide 
into  360  equal  parts,  called  degrees*  Each  degree  is  divided 
into  GO  minutes,  and  each  minute  into  60  seconds. 


Table. 

GO  seconds  (").    .    .    ,     make  1  minute, marked    '. 

GO  minutes 1  degree, °. 

30  degrees 1  sign, 8. 

13  signs,  or  300°, 1  circle, o. 


Table  Reversed. 

/ 

. 

// 

o 

1 

ZiZ 

60 

1      = 

60 

= 

3600 

a 

1     =       30     = 

1800 

= 

108000 

1      = 

=     12     =     3G0     = 

21600 

= 

1296000 

Miscellaneons  Table. 

12  units,  or  things,      .     .     make  1  dozen. 

12  dozen 1  gross. 

12  gross  or  144  dozen,    ....   1  great  gross. 

20  things 1  score. 

100  pounds 1  quintal  of  fish. 

19G  pounds 1  barrel  of  flour. 

200  pounds 1  barrel  of  pork. 

18  inches 1  cubit. 

22  inches,  nearly, 1  sacred  cubit. 

14  pounds  of  iron  or  lead    ...    1  stone, 

21i  stones 1  pig- 

b  i>igs  1  fother. 


420 


APPENDIX. 


BOOKS  AND  PAPER. 


The  terms,  folio^  quarto,  octavo,  duodecimo,  &c.,   indicates 
the  number  of  leaves  into  which  a  sheet  of  paper  is  folded. 


A  sheet  folded  in  2  leaves 
A  sheet  folded  in  4  leaves 
A  sheet  folded  in  8  leaves, 
A  sheet  folded  in  13  leaves 
A  sheet  folded  in  16  leaves 
A  sheet  folded  in  18  leaves, 
A  sheet  folded  in  24  leaves 
A  sheet  folded  in  32  leaves 


24  sheets  of  paper, 
20  quires,     .    .    . 


6  biindleo. 


is  called,  a  folio. 


a  quarto,  or  4to. 

an  octavo,  or  8vo. 

a  12mo. 

a  16mo. 

an  ISmo. 

a  24mo. 

a  82mo. 

make    1  quire. 
...    1  ream. 


1  bundle. 
1  UMe. 


METRIC  SYSTEM  OF  WEIGHTS  AND 
MEASURES. 

The  primary  base,  in  this  system,  for  all  denominations  of 
weights  and  measures,  is  the  one-ten-millionth  part  of  the  dis- 
tance from  the  equator  to  the  pole,  measured  on  the  earth's 
surface.  It  is  called  a  Meter,  and  is  equal  to  39.3t  inches, 
very  nearly. 

The  change  from  the  base,  in  all  the  denominations,  is  accord- 
ing to  the  decimal  scale  of  tens  :  that  is,  the  units  increase  ten 
times,  at  each  step,  in  the  ascending  scale,  and  decrease  ten 
times,  at  each  step,  in  the  descending  scale. 

MEASUKES  OF  LENGTH. 
Base,  1  meter  =  39.37  inches,  nearly. 

Table. 


Ascending  Scale 

• 

1 

Descending  Scale. 

Myriameter. 
Kilometer. 

-I 

o 
o 

w 

c 

1 

Centimeter. 
Millimeter.       J 

1       1 

1 

1 

1 

1 

1      1 

The  names,  in  the  ascending  scale,  are  formed  by  prefixing  to 
the  base,  Meter,  the  words,  Deca  (ten),  Hecto  (one  hundred), 
Kilo  (one  thousand),  Myria  (ten  thousand),  from  the  Greek  nu- 
merals ;  and  in  the  descending  scale,  by  prefixing  Deci  (tenth), 
Centi  (hundredth),  Milli  (thousandth),  from  the  Latin  numerals. 


4:23  METIUC  SYSTEM. 

Hence,  the  name  of  a  unit  indicates  whether  it  is  greater  or  less 

than  the  standard  ;   and,   also,  how  many  times.     The  table  is 

thus  read : 

10  millimeters      make       1  centimeter. 

10  centimeters  make  1  decimeter. 

10  decimeters  make  1  meter. 

10  METERS  make  1  decameter. 

10  decameters  make  1  hectometer. 

10  hectometers  make  1  kilometer. 

10  kilometers  make  1  myriameter. 


Table  of  Equivalents. 


i 


.-a|i§  a  I  a 

&flH(Sl                     ft                       I  I 

1=  10 

1=            10=  100 

1=           10=           100=  1,000 

1=       10=         100=        1,000=  10,000 

1=       10=     100=      1,000=      10,000=  100,000 

1=   10=     100=  1,000=  10,000=    100,000=  1,000,000 
1  =  10=100  =  1,000  =  10,000  =  100,000  =  1,000,000=10,000,000 

Table  of  Equivalents  in  English  Measure. 

1  Millimeter  =  0.0394  inches,  nearly. 

1  Centimeter  =  0.393t     " 

1  Decimeter    =  3.9310     " 

1  Meter         =  39.31  in.  =  3.280833  ft. 

1  Decameter  =  32  ft.  9.1  in. 

1  Hectometer  =  19  rd.  14  ft.  1  in. 

1  Kilometer    =  4  fur.  38  rd.  13  ft.  10  in. 

1  Myriameter  =  6  mi.  1  fur.  28  rd.  6  ft.  4  in. 

Besides   a  clear   ajDprehension    of   the    length    of   the    base, 
1  meter,  it  is  well  to  consider  the  length  of  the  largest  unit,  tho 


MEASURES  OF  LENGTH.  423 

myriameter,  equal  to  nearly  6  and  one-fourth  miles  ;  and  also  tho 
length  of  the  smallest  unit,  the  millimeter,  about  four-hundredths 
of  an  inch.  Compare,  also,  each  of  the  smaller  measures,  the 
decimeter  and  centimeter,  with  the  inch. 

When,  in  the  metric  S3'stem,  the  value  of  any  single  unit 
is  fixed  in  the  mind,  the  values  of  all  the  others  may  be  readily 
apprehended,  since  they  always  arise  from  multiplying  or  dividing 
by  10. 

Note. — In  all  the  tables,  tlie  unit  is  in  small  cax)itals,  and  should  bo 
constantly  referred  to. 

Methods  of  Reading. 

The  number  25365.891  meters,  is  read,  in  English, 

Twenty-five  thousand  three  hundred  and  sixty-five  meters,  and 
897  thousandths  of  a  meter.  But  in  the  language  of  the  metric 
system,  it  may  be  read, 

Two  myriamcters,  5  kilometers,  3  hectometers,  6  decameters, 
5  meters,  8  decimeters,  9  centimeters,  and  7  millimeters.  It  may 
also  be  read,  beginning  with  the  lowest  denomination,  7  milli- 
meters, 9  centimeters,  &c.,  &c. 

In  reading,  remember  that  the  unit  of  any  place  is  ten  thnes 
as  great  as  the  unit  of  the  place  next  at  the  right,  and  one- 
tenth  of  the  unit  of  the  place  next  at  the  left.  Hence,  the 
change  from  one  unit  to  another,  and  the  methods  of  reduction 
and  reading,  are  identical  with  those  in  the  system  of  decimal 
currency. 

1.  Write,  numerate,  and  read,  five  hundred  and  ninety-six 
hectometers. 

2.  Write,  numerate,  and  read,  eighty-nine  thousand  and  forty- 
one  centimeters. 


Questions. — VVHiat  is  tlio  primary  base  of  the  metric  system?  To 
■what  portion  of  the  earth's  surface  is  it  equal?  What  is  its  length? 
What  is  the  ascending  scale  from  the  meter?  What  is  the  descending 
scale  ?  What  is  the  length  of  a  myriameter  ?  According  to  what  law 
do  the  different  units  increase  and  decrease  ? 


424:  METEIC  SYSTEM. 


MEASURES  OF  SURFACES,  OR  SQUARE  MEASURE. 

Base,  1  Are  =  the  square  whose  side  is  10  meters. 
=  119.6  square  yards,  nearly. 
=  4  perches  or  square  rods,  nearly. 

The  unit  of  surface  is  a  square  whose  side  is  10  meters.     It  is 
called  an  Are,  and  is  equal  to  100  square  meters. 

Table. 


1 

1 

!2; 

< 
1 

1 

6 
1 

The  table  is  thus  read  : 

100  centares 

make 

1   ARE. 

100  ares 

make 

1  hectare. 

Table  of  Equivalents. 

Hectare. 

Akb. 

Centare. 

, 

1   = 

100 

1  =  100  =  10,000 

Equivalents  in  acres,  roods,  and  perches. 

1  Centare  =  1.195985  sq.  yards,  nearly. 
1  Are        =  3.9536t  perches. 
1  Hectare  =  2A.  IR.  35.367P. 

MEASURES  OF  VOLUMES. 
Base,  1  liter  =  the  cube  of-  the  decimeter. 
=  61.023378  cubic  inches. 
=  a  little  more  than  a  wine  quart. 

Questions. — ^What  is  the  primary  base  of  the  measure  for  surfaces  ? 
To  what  is  it  equal,  in  square  yards  ?  What  are  the  denominations,  be- 
ginning with  the  least  ?  To  what  is  the  centare  equal  ?  To  what  is  the 
hectare  equal  ? 


MEASURES   OF   VOLUMES.  4:2o 

The  unit  for  the  measure  of  vohime  is  the  cube  whose  edge 
is  one-tenth  of  the  meter — that  is,  a  cube  whose  edge  is  3.93t 
inciies.  This  cube  is  called  a  Liter,  and  is  one-thousandth  part 
of  the  cube  constructed  on  the  meter,  as  an  edge. 


The 


Table. 

Ascending  Scale.  Descending  Scale. 


' 

' 

' 

2 

3 

OQ 

B 

55 

55 

o 

'3 

Decaliter. 
Liter.    U 

1 

u 

S 

1 

1 

1 

1 

1       1 

1 

1 

1 

:>\q  is  thus 

read 

: 

10  milliliters 

make 

centiliter. 

10  centiliters 

make 

deciliter. 

10  deciliters 

make 

liter 

10  liters 

make 

decaliter. 

10  decaliters 

make 

hectoliter. 

10  hectoliters 

make 

kilolitcr,  or  stere, 

Table  of  Equivalents 


I    1 

a      1 


1  =  10  = 
=  10  =  100  = 


I 

1  = 

10  = 
100  = 


1  = 

1  =    10  = 

10  =   100  = 

100  =  1,000  =  10,000  = 

1,000  =  10,000  =  lOOjOOO  = 


1,000  = 


10 

100 

1,000 

10,000 

100,000 

1,000,000 


NoTK— The  kiloliter,  or  stere,  is  the  cube  constructed  on  the  meter, 
uf;  an  edge.    Honce,  the  liter  is  one-thousandth  part  of  the  kiloliter. 


426  METRIC   SYSTEM. 

Equivalents  in  Cubic  Measure. 
1  milliliter  =  .061023  cubic  inches. 

1  centiliter  =  .610234  cubic  inches. 

1  deciliter  =  6.102338  cubic  inches. 

1  LITER  =         61.023378  cubic  inches. 

1  decaliter  =       610.233179  cubic  inches. 

1  hectoliter  =    6102.337795  cu.  in.  =    3.5314454  cu.  ft. 

1  kiloliter,  or  stere  =  61023.377953  cu.  in.  =  35.314454    cu.  ft. 

Note. — Law  of  change  in  the  units,  and  methods  of  reading,  are  the 
same  as  in  linear  measure. 

DRY  MEASURE. 

EQUIVALENTS    IN   THE   WINCHESTER   BUSHEL. 

Since  1  bushel  =  2150.4  cu.  in. ;  1  pk.  =  537.6  cu.  in.  ;  1  qt.= 
67.2  cu.  in  ;  1  pt.  =  33.6  cu.  in.  ;  therefore, 
=  .001816  pints. 
=  .018161  pints. 
=  .181611  pints. 
=  1.816112  pints. 
=  1  pk.  1.08056  qt. 
=  2bu.  3pk.  2qt.  1.6112  pt. 
=  28  bu.  Ipk.  4qt.  0.112pt. 

aid,  is  a  little  less  than  1  quart,  and  the 
stere,  nearly  30  Winchester  bushels. 

LIQUID  MEASURE. 

EQUIVALENTS   IN   THE   WINE    GALLOIT. 

Since  1  wine  gallon  contains  231  cubic  inches,  1  quart  will 
contain  57.75  cubic  inches;  1  pint,  28.875  cubic  inches;  and 
1  gill,  7.21875  cubic  inches  ;    we  have, 

Questions. — What  is  tbe  unit  for  the  measure  of  volumes  ?  To  wbat  is 
it  equal  in  cubic  inches  ?  What  part  is  it  of  the  cube  on  the  meter? 
Name  all  the  denominations  of  volume.  What  is  the  unit  of  Dry  Meas- 
ure ?    To  what  is  it  equal  ?    To  what  is  the  stere  or  kiloliter  equal? 


milliliter 

centiliter 

deciliter 

LITER 

decaliter 

hectoliter 

kiloliter,  or 

stere 

Note.— 

-The  liter,  or 

stand 

milliliter 
centiliter 
deciliter 

LITER 

decaliter 
hectoliter 


WEIGHTS. 

=  0.008453  gills. 

=     .084534  gills. 

=    .845345  gills. 

=  Iqt.  .11336  pt. 

=  2  gal.  2qt.  1  pt.  .1330  pt. 

=  26 gal.  Iqt.  1  pt.  1.344  gills. 


427 


1  kiloliter,  or  stere  =  1  tun,  12  gal.  0  qt.  1  pt.  1.44  gilis. 

WEIGHTS. 
Base,  1  gram  =  weight  of  a  cubic  centimeter  of  rain-water. 
=  15.432  grains,  Troy,  nearly. 
=  .0352746  ounces,  Avoirdupois,  nearly. 

The  unit  of  weight  is  also  equal  to  the  one-millionth  part  of 
the  weight  of  a  cubic  meter  of  pure  rain-water,  weighed  in  va- 
cuum.  It  is  called  a  Gram,  and  is  equal  to  15.432  grains,  Troy, 
which  is  equal  to  .0352746  ounces,  Avoirdupois,  very  nearly. 

Table. 


Ascending  Scale. 

Descending  Scale. 

'o 

&i 

[Ulier,  tonn 
uintal. 
yriagram. 
ilogram. 

ectogram. 
ecagram. 

!  1 1 1 

^     C     ^     M 

W     P 

C!5      ft      6      S 

1111 

1      1 

;     1     1     1 

The  table  is  thus  read  : 

10  milligrams 

make 

I  centigram. 

10  centigrams 

make 

1  decigram. 

10  decigrams 

make 

1  gram. 

10  grams 

make 

1  decagram. 

10  decagrams 

make 

1  hectogram. 

10  hectograms 

make 

1  kilogram. 

10  kilograms 

make 

1  myriagram. 

10  myriagrams 

make 

1  quintal. 

10  quintals 

make 

1  millier,  or  touHcau. 

428 


METRIC    SYSTEM. 


i 

Table  of  Equivalents. 

1  1 

Myriagram. 
Kilogram. 

g 

2 

Decagram. 
Gram. 

Decigram. 
Centigram. 

Milligram. 

, 

1=                10 

. 

1=              10=               100 

. 

1=             10=            100=            1,000 

. 

1=           10=            100=          1,000=           10,000 

1= 

10=         100=         1,000=        10,000=         100,000 

1 

10= 

100=       1,000=       10,000=       100,000=       1,000,000 

1=     10= 

100= 

1,000=     10,000=     100,000=    1,000,000=     10,000,000 

.    1= 

10=   100= 

1,000= 

10,000=  100,000=  1,000,000=  10,000,000=  100,000,000 

1=10= 

100=1,000=10,000= 

100,000=1,000,000=10,000,000=100,000,000=1,000,000,000 

Equivalents  in  Avoirdupois  and  Troy  Weights. 


1  Milligram 

= 

0.0154  grains,  Troy. 

1  Centigram 

= 

0.1543  grains, 

It 

1  Decigram 

= 

1.5432  grains, 

It 

1  Gram 

= 

15.4327  grains, 

It 

1  Decagram 

= 

0.352t  ounces, 

Avoirdupois, 

1  Hectogram 

= 

3.5274  ounces, 

(( 

1  Kilogram 

= 

2.2046  pounds, 

(( 

1  Myriagram 

= 

22.046    pounds, 

(( 

1  Quintal 

= 

220.46      pounds. 

ti 

1  Millier,  or  ton.  =2204.6        pounds,         *' 

Note. — Law  of  change  in  the  units,  and  methods  of  reading,  the  same 
as  in  linear  Measure. 


NATURE  OF  THE  METRIC  SYSTEM. 

The  Metric  system  is  based  on  the  meter.  From  the  meter, 
three  other  units  are  derived ;  and  the  four  constitute  the 
primary  units  of  the  system.     They  are  : 


Questions. — What  is  the  unit  of  weight  ?  To  what  is  it  equal  in  Troy 
weight  ?  To  what  is  it  equal  in  Avoirdupois  ?  Name  aU  the  units  of 
the  weight,  from  the  lowest  to  the  highest.  To  what  is  the  millier,  or 
ton,  equal  ? 


GENERAL  PRINCIPLES. 


429 


Meter  =  39.3Y  inches,  nearly  :  unit  of  length. 
Are      =  a  square  on  10  meters :  unit  of  surface. 
Liter    =  a  cube  whose  edge  is  a  dociiiieter  :  unit  of  volume. 
Gram    =  the   weight  of  a   cube   of   rain-water,   each  edge  of 
which  is  a  centimeter :  unit  of  weight. 

From  these  four  units  all  others  are  derived,  according  to 
the  decimal  scale. 

Every  system  of  Weights  and  Measures  must  have  an  inva- 
viable  unit  for  its  base  ;  and  every  other  unit  of  the  entire 
system  should  be  derived  from  it,  according  to  a  fixed  law. 

Tlie  French  Government,  in  order  to  obtain  an  invariable 
unit,  measured  a  degree  of  the  arc  of  a  meridian  on  the  earth^s 
surface  ;  and  from  this  computed  the  length  of  the  meridional 
arc  from  the  equator  to  the  pole.  This  length  they  divided  into 
ten  million  equal  parts,  and  then  took  one  of  these  parts  for 
the  unit  of  length,  and  called  it  a  Meter.  The  length  of  this 
meter  is  equal  to  1  yard,  3  inches,  and  37  hundredths  of  an  inch, 
very  nearly.  Thus  they  obtained  the  length  of  the  unit  which 
is  the  base  of  the  Metric  System  of  Weights  and  Measures. 

The  next  step  was  to  fix  the  law^  by  which  the  other  units 
should  be  obtained  from  the  base.  The  scale  of  tens  was 
adopted. 

rRONUNCIATION. 


Me'teb. 

Mil'li-me-ter. 

Cen'ti-me-ter. 

Dcc'i-me-ter. 

Dee'a-me-ter. 

Hec'to-me-ter. 

Kiro-me-ter. 

Myr'i-a-me-ter. 


Abe. 


Cen'tare. 


Hee'tare. 


Li'tee. 

MQ'li-li-ter. 

Cen'ti-li-ter. 

Dec'i-li-ter. 

Dee'a-li-ter. 

Hee'to-li-ter. 

Kiro-li-ter. 

Myrl-a-li-ter. 


Geam. 

Milli-gram. 

Cen'ti-gram. 

Dec'i-gram. 

Dee'a-gram. 

Hee'to-gram. 

Kil'o-gram. 

Myr'i-a-gram. 


430  METRIC    SYSTEM. 


TO  CHANaE  FKOM  ONE  SYSTEM  TO  THE  OTHER. 

To  change,  in  Linear  Measure,  from  the  Metric  to  the  Common 
system. 

Rule. 

Multiply  the  meters  and  decimals  of  a  meter  by  3.280833 

(the  value  of  a  meter),  and  the  product  will  he  the  result  in 

feet. 

To  change  from  the  Common  to  the  Metric  system. 

Rule. 
Beduce  the  linear  measure  to  feet  and  decimals  of  a  foot^ 
and  then  divide  by  3.280833  ;  the  quotient  will  be  the  result  in 
meters  and  decimals  of  a  meter. 

Examples. 

1.  In  5961.814  meters,  how  many  feet  and  inches? 

2.  In  814163  meters,  and  31  hectometers,  how  many  feet  and 
inches  ? 

3.  Express   320  rods,   5   yards  and   6  inches   in  the  Metric 
Measures. 

4.  Express  I  mile,  3   furlongs,   39   rods  and  5  yards  in  the 
Metric  Measures. 

To  change,  in  Square  Measure,  from  the  Metric  to  the  Common 

system. 

Rule. 

Beduce  the  number  to  ares  and  decimals  of  the  are;  then 

multiply  by   3.95361,    and  the  product  will  be  the  residt  in 

perches. 

To  change,  from  the  Common  system,  to  the  Metric  system. 

Rule. 
Find  the  value  of  the  number  in  perches  and  decimals  of 
a  perch :  then  divide  by  3.95361,  and  the  quotient  tvill  he  the 
result  in  ares  and  decimals  of  the  are. 


REDUCTION.  431 

Examples. 

1.  In  6127  ares,  4  liectares  and  3  centares,  bow  many  acres, 
roods  and  perclies? 

2.  In   32t   ares,   15    hectares    and   89   centares,  how    many 
>  square  feet? 

^      3.  In  4   acres,   3   perches   and   200   square  feet,   how  many 
hectares,  ares  and  centares  ? 

4.  In   1375   square   yards    and   250   square   feet,   how  many 
hectares,  ares  and  centares? 


To  change,  in  measures  of  volume,  from  the  Metric  to  the  Com- 
mon system. 

Rule. 
Reduce  the  number  to  liters  and  decimals  of  the  liter  :  then 
multiply  by  Gl. 023378,  and  the  product  will  be  the  result  in 
cubic  inches. 


To  change,  in  measures  of  volume,  from  the  Common  to  the 
Metric  system. 

Rule. 

Reduce  the  number  to  cubic  inches :  then  divide  by  61.023378, 
and  the  quotient  will  be  the  result  in  litebs  and  decimals  of 
the  liter. 

Examples. 

1.  In  6  kiloliters,  9  hectoliters,  6  decaliters,  8  liters  and  4 
centiliters,  how  many  cubic  feet  and  inches  ? 

2.  In  8  hectoliters,  9  decaliters,  27  liters  and  15  milliers, 
how  many  cubic  yards,  feet  and  inches  ? 

3.  Change  27  cubic  yards,  16  cubic  feet  and  16  cubic  inches, 
to  the  Metric  measures. 

4.  Change  40  cubic  yards,  25  cubic  feet  and  1167  inches,  to 
the  Metric  measures. 


432  METEIC   SYSTEM. 

To  change,  in  weights,  from  the  Metric  to  the  Common  system. 

Rule. 
Beduce  the  number  to  grams  and  decimals  of  a  gram :  then 
multiply  by  15.423,  and  the  product  will  be  the  result  in  grains 
Troy;   or,  multiply  by   .0352146,   and  the    product  will  be 
ounces  in  Avoirdupois. 

To  change,  in  weights,  from  the  Common  to  the  Metric  system. 

Rule. 
Beduce  the  number  to  Troy  grains,  or  to  Avoirdupois  ounces: 
then  divide  by  15.423,  or  by  .0352746,  and  the  quotient  mil  be 
GRAMS  and  decimals  of  the  gram. 

Examples. 

1.  Change  4  quintals,  6  kilograms,  4  decagrams,  T  grams  and 
6  centigrams,  to  Avoirdupois  and  Troy  weights. 

2.  Change  2  milliers,  6  myriagrams,  9  grams,  4  decagrams  and 
9  milligrams,  to  Troy  and  Avoirdupois. 

3.  Change  1  T.  3  cwt.  3  qr.  20  lb.  6  oz.,  to  the  Metric  weights. 

4.  Change  161b.  11  oz.  4  pwt.  19gr,,  Troy,  to  the  Metric 
weights. 

Ques. — In  linear  measure,  how  do  you  change  from  the  Metric  to  the 
Common  system  ?  How  do  you  change  from  the  Common  to  the  Metric 
system  ? 

In  square  measure,  how  do  you  change  from  the  Metric  to  the  Common 
system  ?    How  do  you  change  from  the  Common  to  the  Metric  system  ? 

In  measures  of  volume,  how  do  you  change  from  the  Metric  to  the 
Common  system  ?  How  do  you  change  from  the  Common  to  the  Metric 
system  ? 

In  weights,  how  do  you  change  from  the  Metric  to  the  Common  sys- 
tem ?    How  do  you  change  from  the  Common  to  the  Metric  system  ? 


ANSWERS. 


PACK.    EX.       AN3.  EX.  ANS.  EX.         ANS. 


16. 
16. 


1  I  J«.  II  2  I  XIY.  II  3  I  XYI.  II  4  I  XYII.  ||  5  |  XIX 


6  I  XXII.  II  t  I  XXYIII.  II  8  I   XXIX.  II  9  |  XXXIII. 


16. 


10  I  XXXYIL  II  11  I  XXXYIIL  ||  12   |  XLIII.  ||  13  | 


XLYII.  II  14  I  XLIX.  II  15  I  LYL||  16  |  LYIII.  ||  17  |  LIX. 


li).  I 


16. 
16 
16. 
16 
16 


18  I  LXY.  II  19  I  LXIX.  II  20  |  LXYII.  ||  21  |  LXXY 
52"7 LXXYI.  II  23  I  LXXXI.  ||  24  |  LXXXYII.  ||  25  | 
LXXXIX.  II   26   I  XCIY.  ||  27  |  XCY.  ||  28   |  XCYIL 


29  I   XCIX.  I  30  I   CXY.  ||  31    |  DCCL.  ||  32   |  MLX. 
33  I  MMXL.  II  34  |  DLX.  ||  35  |  DCCCCLX.  ||  36  |  DCXC. 


a7  I  ML.  li  38  I  MMMMIY.  ||  39   |  YMIX.  ||  40  |  IXIX. 


16. 


41  I  DCCCYI.  II  42  I  DCYIII.  j  43  |  YMMMYI.  ||  44  | 


16. 


MMI.    II    19.    II    1    I    7    II    2    I    80   II   3   I   9000  ||  4  |   93 


W. 


5  I  961  II  6  I  7408  ||  7  |  897021  ||  8  |  86029430  ||  9  |  4328- 


20. 


I  021063  II  10  I  967040932  i|  11  |  30430208123  ||  12  |  360- 


20. 


030702010  II  13  I  5800006000812  ||  14  |  75605070905008 


20, 


I  15  I  904000800200720  ||  16  |  6000900704098020  ||  17  | 


20. 


80510006040900040900  ||  18  |  6050900001  ||  21.  ||  19 


21 


987054321012345678  ||  22.  ||  1  |  621  ||  2  |  5702  ||  3  |  8001 


22. 


4  I  10406  II  5  I  65029  ||  6  |  40000241  ||  7  |  59000310 


22. 


I  8     I     12111.11    9     I     300001006    ||.10     |     69003000200 


23. 


32  I  47000069000465207  ||  33  |  800000000000429006009 


434  ANSWERS. 


•23.  II  34  I  95000000000000089089306  ||  35  |  6000000451065- 
23.  II  047104  11  36  1  999065841411  H  30.  1|  1  |  2  ;  1  ||  2  |  t  ;  3 
30.  II  3  I  1  ;  t  II  32.  1|  6  |  42600  ;  426000  ||  1  \  36860  |1  8  |  $8.75 
32.  II  9  I  433005  \\  10  |  8996  i  11  1  £1  12s.  8d.  1  far.  H 12  |  154451b. 

32.  II  13   I    IT.   Ucwt.  Iqr.   201b    ||    33.    |1    14    |    26215  grs 

33.  li  15  I  1221b.  2oz.  18pwt.  9gr.  |!  16  |  29362gr.  ||  17  | 
33.  II  301b.  4  §  3  3  2^  7gr.  |1 18  [  249  in.|l  19  |  1600rd.  8800yd.; 
33.  II  26400  ft.  316800in.  ||  20  j  75yd.  2ft.  Gin.  ||  21  | 
33.  II  6  sq.  yd.  2  sq.  ft.  ||  22  |  2  A.  OR.  35 P.  ||  23  |  45  A.  6sq.  Ch. 
33.  II  24  1  568  P.  ||  25  |  967680  cu.  in.  |1  26  1  3968  ou.  ft. 
33.  II  27  I  440  cords  ||  28  1  2512  na.  U  29  |  144  yd.  1|  30  | 
33.  II  78  E.  E.  1  qr.  ||  31  1  1008  qt.  ||  32  1  15  hlid.  ||  33  1  3024  pt. 
33.  II  34  1  129  bar.  (  35  |  1984  pt.  |  36  |  32  bu.  3pk.  7  qt. 
33.  II  37  I  63113856  sec.  ||  38  |  8mo. '2wk.  ||  37.  ||  1  |  182630 
37.  II  2  I  87539  ||  3  |  110526  ||  4  |  79165  1  5  |  73285  ||  6  |  4148- 
37.  II  907   II  7  I  395873  ||  8  j  24177  ||   9  |  66395   |j   10  |  22099 

37.  II  11  I  73566  ||  12  |  833157  ||  38. -||  13  |  32921  ||  14  |  185876, 

38.  II  15  I  93684  ||  16  |  34289  |1  17  |  243972  1  18  |  $991,546 
38.  II  19  I  $85,465  ||  20  I  $770,560  1|  21  1  525.892  (  22  |  $9638.495 
38.  II  23  I  ie223  2s.  5d.  Ifar.  ||  24  |  12961b.  10  oz.  2pwt. 
38.  II  25  1  453  ft  9  5  3  3  ||  26  |  2  cwt.  3  qr.  8  lb.  8  oz.  5  dr. 
as.  II  27    1    43  T.  2  cwt.  Oqr.    71b.    ||    28    |    312  yd.   Oqr.  2iia. 

38.  II  29   I   251    E.   E.    Iqr.    3  na.    ||    30    |    143  L.    2  mi.    6  fur. 

39.  II  31  I  4  fur.  Ird.  4  yd.  Oft.  7  in.  ||  32  .|  322  A.  IK  4  P. 
39.  II  33     I     2224  Tun    Ohhd.    5  gal.      ||      34    |    339  gal.    3  qt. 


ANSWERS.  435 


39. 

|35 

1   230  chal.  25  bu.    3  pk.  4  qt.  ||  36  |  820  yr.  4  mo.  5  da. 

39. 

|37 

1  904  da.  18  hr.  Imi.  ||  38  |  2T.  14cwt.  Iqr.  201b.  15oz. 

39. 

1  39 

1     23592550    ||    40     |    $137915940    J    41     |    88056 

39. 

|42 

1    121  mi.   4  fur.    8rd.   5  ft.    ||    40.    1    43    |    $22,009 

10.  1 

|44 

1    $27,740     1     45    1    2 Tun  2hhd.  29 gal.  2qt.  Opt 

40. 

|46 

1    $20308675    [    47    |    $30569853    J    48    |    $29026 

40. 

|49 

1  $8209.75  II  50  |  $150106  ||  51  |  29714  |1  41.  ||  52  1  $50- 

41. 

1  110025    1    53  1   59808512    ||    54   |   2T.  4cwt.  2  qr.  lib. 

41. 

|55 

1    205  acres.    ||    56    |    $75002.295    |    57     |    $7425 

41. 

1  58 

1  41b.  5oz.  6pwt.  II  59  1  1053420  1  42.  ||  60  |  1842yrs. 

42. 

|61| 

32341  II  62  1  $27131.23  ||  63  |  $28,105  ||  64  |  39yd.  Iqr. 

42. 

1  G5 

1  $180,825  II  66  1  $35068.807  |  67  |  £59  2s.  3d.  2  far. 

42. 

1  68 

1    66585383    ||   43.    ||    69  |   $1019.10  ||  70   |   $33800 

43. 

|71 

380  bu.  1  pk.  II  72  1  $458,342  ||  73  |  £51  14s.  2d.  3  far. 

43.  1 

174 

1   $6235  II   75  1  66°  50'   ||   76  |  10  cents.  |   77  |  5860 

47.  1 

|1    1 

363296   1  2   |  56579  ||    3   |    733071    1    4   |   1711927 

47.  1 

5  1 

41923288   ||   6    |   7838180   ||   7    |    106026   ||   8  |  4391 

47.  1 

9    1 

62786    II    10     1     198621115    ||     11     |    3591757651 

48.  1 

12 

4199675    II    13  |  8878778    ||    14    |   99999977    ||    15  | 

48.  1 

1  88443.641     5     16    |    $806,384     |     It    |    $4853673.758 

48.  1 

18 

1    £U  18s.  3d.  Ifar.    ||    19    |    3T.  8cwt.  2qr.  71b. 

48.  1 

1  20 

1    117yd.  2qr.  Ina.   ||    21    |    59 L.  Imi.  3 fur.  28 rd. 

48.  1 

22 

1    8  Tun  Ihhd.  53gal.  3qt.    ||    23   |    89  A.  2  R.  37  P. 

48.1 

24 

1   975  bu.  Ipk.  6qt.    ||    25    |    124  cords  58  ft.  522  in. 

48.  1 

126 

1   25  E.   E.    Iqr.   3na.     [      27    [    79ft>    10 1    6  3 

43G  ANSWERS. 


28  1  123  43  23  II  29  |  124  E.  E.  3qr.  3  na 
30  I  96  E.  F  Iqr.  1  ua.  \\  31  j  12  T.  lUwt.  3qr 
32  I  2cwt.  2qr.  221b.  ||  33  j  69  qr.  21b.  14  oz. 
34  I  1341b.  14  oz.  13  dr.  ||  49.  ||  35  j  10  A.  2R.  18  P. 
36    I    3tA.  2R.  34P.      ||      3T    |    14t  da.  21  hr.  66  mm 


38  I  52  hr.  50mm.  54sec.  ||  39  |  $8759.625  I  40  |  183666662 
41  I  6yr.  9mb.  3wk.  Ida.  ||  42  |  88ft)  0|  63 
43  I  $8.20  II  44  I  $39,868  ||  45  |  $10,626  ||  46  | 
£121  17s.  Od.  Ifar.  |  47  |  6yr.  Omo.  Owk.  6da.  9hr.  2mm. 
48  I  6353870  ||  49  |  5747  ||  50  |  $6020  ||  51  |  25712808.91 
52  I  36190  II  53  |  683021  ||  54  |  107445034  ||  55  |  6274 
56  I  4T.  3cwt.  2qr.  231b.  ||  57  |  £19  19s.  2d.  3 far. 
58  I  2299  mi.  2  fur.  4  rd.  ||  59  |  $199,625  ||  60  |  $175,875 
61  I  $3.25  II  61.  II  62  |  19987563  ||  63  |  2899248 
64  I  $73675  ||  65  |  22815  ||  66  |  $198,625  ||  67  | 
80  yr.  8  mo.  Oda.  3hr.  30  min.  ||  68  |  655.125 
69  I  249yr.  Imo.  llda.  ||  70  |  17877  ||  71  |  $7310756 
72  I  4cwt.  Iqr.  181b.  ||  73  |  7398  ||  74  |  2360  ||  75  |  $526 
76  I  6274  II  77  |  $356.35  gain.  ||  78  |  3  A.  2K  39  P. 
79  I  41  cords  5  cord  ft.  ||  80  |  $3280.105  ||  81  |  $44161.987 
82  I  2yr.  8  mo.  19  da.  ||  53.  ||  83  |  $14352.50  ||  84  | 
30  gal.  2  qt.  1  pt.  ||  85  |  50062  ||  86  j  15550  ||  87  1 12°  23'  53' 
88  I  $161,175  loss.  II  89  |  2271707  ||  90  |  32  yd.  Oqr.  2ua 
91  I  £950  2s.  8d.  ||  60.  ||  1  |  6776368  ||  2  |  68653214 
3  I  3422454  ||  4  |  1952883  ||  5  |  4354224  ||  6  j  1028540646 


ANSWERS.  437 


7  I  246G8698404  ||  8  |  3329480  ||  9  |  4036084764 
10  i  129844534245  ||  61.  ||  U  |  810444  ||  12  |  23()13 
13  I  72127422  ||  14  |  5403312  ||  15  |  12440'.)7 
16  I  1990170000  II  17  |  3165172200  1|  18  |  582400000000 
19  I  $104448.48  |  20  |  $2501.136  ||  21  |  $23121.312 


22  I  $71997.312  |j  23  |  $7019.168  |  24  |  $30780.960 
25  I  $21597.440  ||  26  |  $38824.056  ||  27 "  |  $278879.3(i4 
28  I  $379255.968  ||  29  |  $9282001.666  ||  30  j  £Sl  6s.  8d. 
31  I  24  T.  7c\vt.  3qr.  ||  32  |  118  yd.  1  ft.  3  in. 
33  I  114°  26'  15"  II  34  |  561ihd.  7  gal.  2qt.  Opt. 
35  I  698  E.F.  ||  62.  ||  1  |  865T.  llcwt.  3qr.  201b. 
2  I  320  JT.  2  mo.  Owk.  Ida.  15  hr.  12  rain.  ||  3  |  4896 
4  I  670460  ;  6704600  ||  5  |  5704900 ;  57049000 
6  I  4980496000  ;  49804960000  ||  7  |  9072040000  ; 
907204000000  ||  8  |  74040900  ;  740409000  ||  9  |  67493600  ; 
67493600000  ||  10  |  129359360000  ||  11  |  13729103000000 
12  I  664763206000000  ||  13  |  8799238229600000 
14  I  2526426017908695000000  ||  15  |  1093689368445084- 
378777040  ||  16  [  16714410677359581583737  1  17  |  $61975 
18  I  3240  I  19  I  2097  ||  20  |  133  yd.  3qr.  2na. 
21  I  £^  19s.  4d.  2  far.  ||  22  |  $1031.68  |  63.  ||  23  |  $15 
24  I  $506.88  II  25  |  $6336  ||  26  |  $5545  ||  27  |  $16763832 
28  I  496  mi.  1  fur.  24  rd.  ||  29  |  $657  ||  30  |  $24,375 
31  I  868  miles  |  32  |  7lb  2  3  7.3  0^  12gr.  j  33  | 
411  l)u.  Ipk.  Oqt.  II  34  I  427816  ||  64.  ||  35  |  $84.26 


4ZS  ANSWERS. 


64.  II  36  I  $168t5.60  (|  3T  |  2T.  18  cwt.  1  qr.  21  lb.  J  38  |  $971.04 
64.  II  39  I  461  left ;  $1315  price.  i|  40  |  $1417  ||  41  j  $65962788.15 

64.  II  42  I  750  II  43  j  13500    ||    44    [    $243.00    ||    45   j    11914 

65.  II  46  I  $4770.755  ||  47  |  $61  ||  48  |  1672  |  49  |  286 yr.  9  mo. 
65.  II  50  I  84  rd.  14  ft.  ||  51  |  50  ||  52  |  24  cords.  ||  53  |  $92  gain. 

65.  II  54   I   216    II    55  |  $149.25    ||    56  j  37816     ||    57    |  $34.88 

66.  H  58  I  669  hhd.  40  gal.  2qt.  \\  59  |  13650000  ||  60  |  $202.50 

66.  II  61  I  $21,475    J    62  |  $927.35    ||    67.    ||    63    |    $18844.01 

67.  II  64  I  $132,935  ||  65  |  £115  18s.  6d.  ||  72.  |1  1  1  6579 
72.  ||2  136842  ||  3  |  269368  ||.  4  |  275155  ||  5  |  7948312 
72.  II  6  I  1147187  ||  7  j  72331642  ||  8  |  £lb  19s.  9d. 
72.  II  9  I  4A.  OR.  33  P.  ||  10  |  9  yd.  2qr.  1  na.  1|  11  |  $79.3445 

72.  II  12  |, $209,728  ||  13  |  $66862.18  ||  73.  ||  14  |  15311409^1 
73.11^1237132   ||   16  |  177242    ||    17    |    68    ||     18    |    44670 

73.  U  19  I  27i|  1  20  I  $17.4512  ||  21  |  $3.842j-«^%  ||  22  |  $1,125 
73.  II  23  I  $0,375  ||  24  |  $0.81  ||  25  |  $5.01  ||  26  |  $52.88  ||  27  |  9 
73.  II  28  I  95  J  29  I  $8  ||  30  |  763521  ||  31  |  407294|-?-f  J 
73.  J  32  j  13195133if|f  j  33  |  125139204||if 
73.  II  34  I  269577255882T-Y4V3  II  35  |  14243757 748fffJ:i 
73.  1  36  I  15395919iffiJ  ||  37  |  30001000/y\V3  II  ^^  | 
73.  II  131809655J^|J^  |  39  |  300335575Jf?i-Jf  ||  40  |  9948157- 

73.  1  977/-,VWt  II  41  I  59085714tVt  II  42  |  1258127JfIM 
73.1143  I  119191753j%V4V6-  II  44  |  17A.  311.  7P. 
73.  11  45  I  Ida.  12 hr.  31min.  30  sec.  ||  46  |  35  mi.  Ofur.  29  rd. 
73.  II  47  I  49  gal.  3,^^  q*-  II  48  1  2  bu.  0  pk.  7  qt.  ||  74.  ||  49  |  $25.25 


ANSWERS.  439 


74.  1  50  I  2s.  4d.  J  51  |  22 mi.  1  fur.  8rd.  J  52  |  316A.  IR  35P. 
74.  l  53  I  $2*1.397+  J  54  |  98765  ||  55  |  $11250  ||  5(j  1 148018fA| 
74.  II  57  I  $4.75  |  58   j   $12.50  \\      59  |  757l88yV4 

74.  1  60  I  $1,625  1  61  |  365  days.  |  62  |  800008  ||  63  |  47 

75.  II  64  I  IT.  13cwt.  3qr.  ||  65  |  45  cu.  ft.  995}|  cubic  inches. 
75.  II  66  I  301^1  tons.  |  67  |  4424^?3  J  68  |  59'  lO'Hf 
75.  li  69  I  5doz.  J  70  |  $4.50  ||  71 1  ^£273  7s.  6d.  ]  72  |  41684xVt 

75.  J  73  I  9  11  74  I  $56  ||  75  |  666-J-§|  U  76  |  200000 

76.  }  I  I  7175  II  2  I  4600  \\  3  |  168525  ||  4  1  76850  ||  1  |  2725 
76.  1  2  1  387321  J  3  |  4413^40  ||  4  |  15423  |1  5  1  2674584 

76.  II  6  I  280082  |1  77.  ||  1  |  4800  U  2  |  5950  ||  3  |  185000 

77.  1  4  I  8380225  ||  1  |  55975066f  ||  2  ]  493574Gfif 
77.  1  3  I  355850400  ||  4  |  148072400  j  1  |  7408000 
77.  1  2  I  2199176000  |  3  |  242601500  ||  4  1  17573500 
79.  II  1  I  $142  II  2  I  $17  II  3  I  $14  ||  4  1  835  ]  5  |  $864  1|  6  |  $172 

79.  11  7  I  $120  II  8  I  $90  ||  80.  ||  1  |  $121,615  J  2  1  $67.50 

80.  J  3  I  $737.88  ||  4  |  $496,875  I  5  |  $118.9145  \\  81.  |1  1 1  $3,024 

81.  II  2  I  $12.8915  II  3  1  $5,922  ;  $6.4575  ;  $9,198  ||  4  |  $18.22765 
81.  1  5  I  $736.68468f  ]  6  1  $876,434  ||  7  1  $2423.09925 

81.  1  8  I  $339286.5375  J  82.  ||  1  |  254  ||  2  |  26251^2^0^ 

82.  1  3  I  291147  ]  4  |  2U4:SSj%  \\  5  |  978  J  6  1  954  J  7  I  140848 
82.  II  8  1  2025  y  9  |  39252  ]  10  |  475542  \\  11  |  242172 
82.  1  12  I  484344  ||  13  |  951084  ||  14  |  2250  ]  15  |  48120 
82.  II  16  1  16215  II  17  1  4S645  ||  18  |  144378  y  84.  ||  1  I  387 
84.  1  2  I  1548  1  3  I  532  II  4  I  804  I  5  I  15911  1  6  |  1935 


440  ANSWERS. 


84.  II  1  I  1809  II  8  I  3216    !|    85.    ||    1  |  1322Hf   ||    2  |  n40j|§ 

85.  II  3  I  218|ff§  II  4  I  83253^%%  ||  5  |  2459^1  ||  6  |  md^\%% 

85.  II  1  I  950HfH  II  86.  II  1  I  imUi  II  2  I  146  II  3  I  9lUim 
S6.  II  4  I  158t32ff|gg  II  5  |  2b^%\%%  \\  6  |  224tffM-§ 
"6.  II  1  I  196ff    II    2  I   3inft    II    3  I  61096ff    ||    4   |   mUi 

86.  II  5  I  909511  II  6  |  6992A|  ||  1  \  6150^^0  ||  8  j  40^9,-3^ 
89.  II  1  I  ^631  17s.  6d.  n  2  I  ^2  9s.  5fd.  ||  3  f  iSl  16s.  lO^d. 
89.  II  4  I  ^594.50  ||  5  |  ^£469  5s.  ||  6  |  iE931  ||  1  |  ^58t  5s. 
89.  II  8  I  ie82  10s.  II  9  |  $2.t0  ||  10  |  $555  ||  11  |  $547.50 
89.  II  12  I  13.00  II  13  I  $812.25  ||  14  |  $24,375  ||  15  |  $63.4375 
89.  li  16  I  $315.40  II  17|$469.03||18|^615s.  II  91.  ||  3  |  10°  34'  0" 

91.  II  4  I  35°  11'  0"  II  5  I  13°  23'  0"  ||  92.  ||  1  |  Ihr.  2mm.  8sec.  p.m. 

92.  II  2    I    2  hr.  55  min.  24  sec.  p.  M.      ||      3   |    8  lir.  12  min.  a.  m. 

92.  II  4    I     Ihr.  2mm.  20 sec.  Fast.    ||     93.     ||    1    |    33°  55' W. 

93.  II  2  I  95°  48' W.;  lOhr.  17mm.  48 sec.  p.m.  ||  3  |  23°  45'  22"  W. 

93.  II  4  I  120°  W.  II  5  I  156°  59'  E.  ||  94.  ||  1  |  $128  ||  2  |  2  bu.  1  pk. 

94.  II  3  I  32  II  4  I  463684  ||  5  |  416664§  ||  6  |  57979}Jf 
94.  H  7|  7mo.lwk.4Jd.||8|12yr.||9|6mo.0wk.5d.l41ir.40mm. 

94.  II  10  I  765  II  11  I  $72  ||  12  |  $5  ||  13  |  $812.25  ||  14  |  $147.9375 

95.  II  15  I  £14  14s.  II  16  I  iE166  2s.  8d.  ||  17  |  6d.  ||  18  |  $6.95175 
95.  II  19  I  $8.64  II  20  |  $93  ||  21  |  36  ||  22  |  451b.  6oz.  Upwt. 
95.  II  23  I  50  II  24  |  $2480  gain  ;  $19  per  acre.  ||  25  |  6780  cu.  ft 

95.  II  26  I  $773,395  ||  27  |  $4.2408  ||  28 1  $16.7025  ||  96.  ||  29  |  768C 

96.  II  30  I  lib.  7oz.  12pwt.  11  gr.  ||  31  |  $10  ||  32  j  2bu.  Ipk.  7qt. 
96.  II  33  I  $0.75  II  34  |  104  1|  35  |  16  ||  36  |  52  gal.  1  qt.  ||  37  |  96 


ANSWERS.  441 


9B.  II  38  I  $598281  ||  39  |  31680  ||  97-  ||  40  |  130  ||  41  |  n93%V-, 
97.  II  42  I  11  hr.  4min.  32 sec.  a.m.  ||  43  |  127°  30' 
97.  i  44  I  67°  35'  A's  long.;  9hr.  19mm.  p.  m.  B's  time. 
:  r.  II  45  I  10  cords  7  C.  ft.  15cii.  ft.  ||  46  |  Icwt.  3qr.  9lb.  lOoz, 

I.  jl  47"    1     $164,475     ||.    48    |    282}t.  6  mo.  8da.     ||     49 
\u.  II  6  gal.  2qt.  Opt.  2  gi.    ||    50  |  6°  13  mi.  Ifur.  34  rd.  2  yd. 

97.  II  51  I  1000000  II  98.  ||  52  |  13824  ||  53  |  36100 
98.11  54    I    14  mi.  5  fur.  21  rd.  8  ft.     ||     55    |    10     ||     56  |   3' 

98.  II  57  I  3yd.  1  qr.  3na.  ||  58  |  33  ||  59  |  13209i?^/, 
98.  II  60     I      111.88      II      61    I     lyr.    205da.    17  hr.    15min. 

98.  II  62  I  $10591021.60  ||  99.  ||  63  |  25  yr.  6  mo.   16  da.  9  hr. 

99.  II  64  I  $2478,  Widow's  share  ;  $1239,  Child's  share.  ||  i^o  \ 
99.  II  13068  II  66  |  107°  47';  1  hr.  llmin.  8  sec.  p.  m. 
99.  II  67  I  4hr.  56min.  p.m.;  26°  east  of  New  York. 
99.  II  68  I  48  hr.  ||  69  |  4333fg-S  ||  100.  ||  70  |  $2  ||  71  |  46ilbs. 

100.  II  72  I  14  days.  ||  73  |  28  bar.  6  gal.  ||  74  |  24  bar.  19  gal. 
100.  II  75  1  $85.33J    ||    76  |  llf^  rolls.    ||    77  |  7  mi.  6fKr.  20 rd. 

100.  1  78  I  87501b.  ||  79  |  $18,025  ||  80  |  2500  bbl.  ||  101.  ||  81  | 

101.  II  482bu.  Ipk.  2qt.=  1st;  160  bu.  3pk.  Oqt.  IJpt.  =  2d  ; 
101.  II  321  bu.  2pk.  Iqt.  Of  pt.  =  3d.  ||  82  |  40°  50'  East  ; 
101.  II  35^o°y  II  83  I  $2400  =  Captain's  ;  81000  =  LieutenantV: ; 
101.  II  $600  =  Midshipman's  ;  and  $200  =  Sailor's.  ||  84  |  87°  ao 
101.  11  85  I  9hr.  33min.  14scc.  a.m.||  86  |  lOhr.  54min.  IOscc.a.m 

101.  II  87  I  19°  II  88  I  4800  yd.  ||  89  |  $7410  ||   102.  i|  90  |  514 

102.  II  91  I  2011bu.  II  92  I  1  yr.  338 da.  22 hr.  ||  93  |  72  =  greater; 


442  ANswjr,KS. 


102.  II  26  =  less.  ||  94  |  $5T  =  less  ;  ^$133  =  greater  ||  95  |  HOda. 


102. 

II  96    1    $1.1875     1 

97    1 

$8383^  = 

A' 

s ;    $85201  = 

=  B's; 

102. 

II  $7708i  =  C's.  II 

98  1  $1 

1651.25  :rr 

1st 

;  $11576.25 

=  2d; 

102.  II  $11496.25  =  3d 

;  $11401.25  =  4th 

.  II 

104.  II  1  1  3 

X  3  ; 

104. 

1  2    X    5  ;    2    X    2 

X    3  ; 

2x7; 

2 

X   2    X    2 

X    2; 

104.  i|2x3x3;2x2x2x3;3x3x3;2x2x7 


104. 

II  2 

1 

2 

X  3  X  5 

;      2  X   11; 

2 

X 

2x2 

X 

2 

X    2; 

104. 

12 

X 

2 

X  3  X  3, 

2  X  19  ;  2  X  2 

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3 

X  3  X  5  ; 

104. 

V 

X 

7 

II  105.  II  3 

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2 

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2  X  2  X 

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2 

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105. 

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X 

2 

X  3  X  5  ; 

2  X  2  X  2  X 

2 

X  J 

2X2; 

2  X 

3 

X  11  ; 

105.  ||2x2xl7;      2x5x7;     2x2x2x3x3 


105. 

M 

1  2  X 

2  X  19 

;  2  X  3 

X 

13; 

2  X 

2 

X  2 

X  2  X  5  ; 

105. 

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X  41 

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X  3  X 

^ 

2  X 

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X  2 

X  2  x  11  ; 

105. 

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X  3  X  17  ; 

105. 

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105. 

||2X2X2X59;  2x2x2x2x2x5;  2x2x11x19 

105. 

1  6  1  5X3X7 

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2X2X3X  3x3;  2x5x  H ; 

105. 

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2    X    2    X 

29;     2X2X2X3X5; 

lOo. 

II  5  X  5  X  5  ; 

5X5X5X3X3; 2x2x2x5x3x3 

105. 

M  1  2,  5,  3  II 

2  1  2,  3,  7  1 

3  1  5,  7,  3  II  4  1  2,  3,  7   II  5  1  2 

107. 

t  1  1  32  II  2  1 

3f  II  3  1  14 

II  4  1  48  II  5  1  8f  II  6  1  4f  II   7  1  8 

107. 

II  8  1  A  II  9  1 

6|  II  10  1  27 

II  11  1  9  II  12  1  36  II  108.  II  13  1  46 

108. 

11  U  1  4  II  15 

16J  II  16  1 

8  II  17  1  4711  II  18  1 15  II  19  16210 

108. 

II  20  1  6f  II  21 

1  Hi  II  22 

illj||23|4i||   110.  II   111260 

110 

II  2  1  7200  B  3 

1    1260  II  4 

1  1008  II   5  1  10500    II   6  1  10800 

ANSWKllS.  44;:^ 


110.  I  7   I   540    J    8  I   420    ||    9  |   336    ||    10  |  1176    ||     11  | 
110.  II  144  rods.     16  days  =  A's  time  ;    12  days  =  B's   time 


110.  II  9  days  =  C's  time.    ||     111.   ||    12  |  $1680.     112  at  Uo 


111.  II  105  at  $16  ;  80  at  $21;  70  at  $24)1  13  |  210  bu.    105 bags 


1 1 1 .  II  70  bbls. ;  30  boxes  ;  14  hhds.  ||  14  |  60  days.    A  =  3  times 


111. 

IB 

=  4  times 

C  =  5  times  ;  D  =  6  times.    |    112.  2 

1  18 

112. 

13| 

12 

1  *\ 

5    II    5  1  6   H    6  1   10 

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114. 

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211 

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62  1  6  1  81  II  1 

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114. 

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25  1 

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12  1  10  1  3  i   U  1  122 

per  head.     13  = 

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114. 

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124.11  II;  i  1   2|J||3|||;  A;§;   J;f    ||  4  ||-<  ;  ^  ; 


444  ANSWERS. 


124.  1  f  ; 

f  il5|-!f;M;M;il;lll  e  1 -51;  if ;  A;  A; 

124.  II  i  II 

125.  11  1  1  if^  II  2  1  -W-  I  3  1  if  Ml  4  1  -?/  11  5  1  H^ 

125.  II  6  1 

IJJIA  11   1    1   5404  11  8    1   IJJii  11   9   1   If  Aiii    11     10   1   li«563 

125.11  11 

1  H¥-^    11    1  1  HMl    2  1  m^~    11    126.   il    3  1  !-%« 

126.  II  4  1 

HV-'    11    5  1  af  1  Ji    11    6  1  WiyJ     11    t  1  4i>ic„,.»jJ 

126.8   1 

"^AV-'   1   9  1  ^flF  II   10  1  H«   II   11  1  '-Vs^ 

126.  II  12 

1  '^\W^'  11  13  1  ^ff"  1   14  1  iijas   1  15  1  241. 

126.  II  16 

1  '-¥-'  II   n  1  H¥-'   II  18 1  HU^  II  19  1  '-"tW-' 

126.  11  20 

1  iill^    II     21    1  -LH-F     II     22    1    "1    II     23    1    3381 

127.  II  1  1 

li  H  2  1  12  II  3  1  5i^-A  II  4  1  241S§    II    5  1  9  II  6  1  66j% 

127.  ni 

11214  1  8  1  225''Jr  1  9  1  040^1,  1  10  |  5f  ^  II  H  I  HyVA 

127.  II  12 

1   225    II    13   1    lOl^i    11    14    1   14    II    15  |,  376|ii 

127.  11  16 

1  1073l3\V    1    128.    II    1  1  +   il    2  1  i  1    3  1  J  II  4  II 

* 

128.  II  5  1 

#  1  6  KH  1  1  1  A  II  8  1  ill  9  1  i  11  10  1  V'  =  ^ 

128.  II  11 

It    II     12  1ffi    11     13  IH    1!     14  1AV    II    15  lit! 

128.  II  16 

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128. 1  21 

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129.  11  4  1 

A    11  5  1  A   11  6  1  lli   in  1  1    11    8  1  35f    1    9  1  141 

129.  i  10 

1  8i  1  11  1  :i¥ff  II  12  1  sth  II  13  1  ISfi  II  130.  Ill  1  e 

130.  1  Vt' 

'.  u  11 2 1  m.  in.  ^¥0-,  m  ii  3 1  'ivv,  m  m,  m 

130.  i  4  1 

if.  Ii.  If.  il  «  II  5  1  VaV,  Mg.  us.  Ill  II  6  i  ftf 

130. 11  m 

\  iiiitVt,ij-hi  8 1  n,  fi,  !i.  ¥«'  II 9 1  ¥o^  ii,  w 

ISO.  11  Vj"  11  10  1  %\  V   IPl  1  %',  w    II  12  1  ¥/.  3\.  W, 

ANSVVfcKS.  445 


130.  II  Ji  II  1  I  A.  tV.  a,  }§  II  3  I  if,  A,  it  II  3  I  U.  il  Vif 
130.  II  4  I  W,  ii  W  II  5  I  '#,  Fo,  Vif  II  6  I  fg,  »-»,  ^«,  Vo° 

liol  ^  I  4J.  M.  'tV.  ft  II 8 1  M.  A.  fl.  IS  II  9  I  H.  H.  U-. 
130.  II  H 11  '0  I  VV.  Vo°.  ih  \V  II  132.  II  1 1  m.  m.  h% 

■32.  II  2  I  H,  ^1,  a    I    3  I  fl,  i^,  A    II    4    I    W.  W.  A 

132-  II  5  I  w.  H.  A  II  6  I  %%'.  A.  il  I  -t  I  A'.  W,  A 
'32.  II  8  I  w,  M.  H.  mi  9  I  i°A.  #?.  AV  II 10  I  ¥/■  W. 
132.  II  |g  I  11  I  W.  Vo',  M  II  12  I  U,  '^',  H  II  13  I  V-/. 
132.  IIW.  ij.  A  II  134.  II  1  I  120  far.  |  2  |  1606?- lb. 
134.  1  3  I  7840 min.  ||  4  j  3240 grs.  1  5  |  ^-jtH.  j  6  j  Ayl- 
134.  i  1  I  ,AW°  II  8  I  ^h  '=»'-d-  i  9  I  8«-  ^d. ;  9s.  4d. 
134.  1  10|6fur.  8rd.  4yd.  2ft.  Sin.  ;  6fur.  34  rd.  1yd.  1ft.  Siin 

134.  II  11  I  33  rd.  1yd.  2  ft.  6  in.  ||  12  |  jhe»'"-  II  13  I  tAt, 
134.11  14   I   ^'^    II    1    I   if"i-     II     2   I   jE4H     II     3   I    jUl 

135.  II  4  I  j-gg  II  5  I  ^f  [|  6  I  till  T  I  iflll  8  I  tVA  I  9  I  A 
135.  II  10  I  A°„-  II  136.  II  2  I  f,  A.  H.  U  I  3  I  A'o.  aV»,  il  il 
137.  II  1  I  2igf  II  2  I  3 /A  II  3  I  2§i  I  4  1 1A"A  i  5  |  4H  ||  6  |  4jg 
137.  1  T  I  2A^  1  8  I  9|i  I  9  I  16|3J    ||   10  |  llA  11  H  I  2CU 


137.  1  12  I  10/A  II  13  I  4|f  1  14  I  6,*A  1  15  1*611  ||  16  |  133,\ 


137.  II  n  I  18A  I  18  I  89A  n  19  |  212A  ||  20  1 161|  ||  21 1  65A'5 

137.  II  22  I  10013  II  23  |  891f    ||     24  |  341li  bushels  ;  UllMl 

139.  1  1  I  14Ain.   I    2  I  2da.  14  hr.  30min.    ||    3  |  Icwt.  1  qr. 

_  _ 


446  ANSWEKS. 


139.  I  191b.  4}oz.    II    4  I  2oz.  10  pwt.  12  gr.    ||    5  |  9  cwt.  1  qr. 

139.  II  51b.  8f  oz.    II    6  |  20 bu.  1  pk.  5f  qt.   ||    7  |  3  hhd.  37  gal. 

139.  II  3 qt.    Opt.    Ifgil.      ||      8    |    55  da.   2  hr.    4tmin.    30  sec. 

m  II  9  I  2R.  20 P.  Usq.  ft.  SSg^-sq.  in.    |]    10   |   7  in.    ||    11  | 

139.  H  13s.  lOfd.  ]|  12  ]  7  fur.  2  ft.  9in.  ||  13  |  222  da.  1  hr.  24min 

1 

J  39.  II  14    1    7oz.  7  pwt.   23  gr.     ||      15    |    5  s.  16°  16'  iO^%%-" 

139.  H  16  I  1  yd.  0  qr.  2^  na.  i|  17  |  1  C.  ft  lieu.  ft.  466f  cu.  in. 
139.1118    1    2C.   4C.  ft.    2cu.  ft.     J)     19    |    3  yd,   2qr.   Ofna. 

140.  II  20  I  3A.  2R.  33}?.  ||  21  |  llcwt.  3qr.  211b.  lloz.  l^dr. 

140.  II  22  I  3  fur.  0»rd.  2  ft.  6  in.  ||  141.  ||  1  |  f  ||  2  |  j\  \\  3  |  ^\ 

141.  a  H  If  II  5  I  A  I  6  I  A  in  Tih  II  8  I  35-11  II  9  I  X 
141.  II  10  I  ff  II  11  I  24^,-  II  12  I  Ijh  II  13  I  1  II  14  I  3A 
141.  1  15  I  7f  II  16  I  14f  I  17  I  A  II  18  I  H  II  19  I  ^  II  20  |  6f^ 

141.  i  21  I  81  II  22  I  H  II  23  |  $72  ||  142.  ||  24  |  $f  ||  25  |  18H 

142.  II  26  I  1811  11.27  |  33/^  ||  28  |  22^%  ||  29  |  1^  j  30  |  $18f 
i"42.  II  2  I  tIj  II  3  I  ^i^  II  4  I  3!^  II  5  Uf  0    II    143.    ||    2  |  2^^ 

143.  i  3  I  761  II  4  1  73fi  ||  5  |  6ff  ||  6  |  182^^^ 
144.1  1    I    9oz.  7  pwt.  12 gr.     1|     2    |    7 cwt.  1  qr.  241b.  8  oz. 

144.  II  3  I  29  gal.  3f  qt.  ||  4  |  1  mi.  1  fur.  16rd.  ||  5  |  Is.  3d. 
144.  II  6  I  38'  34f'  ||  7  |  563  A.  OR.  35|-P.  J  8  |  10  cwt.  Iqr. 
144.  II  221b.  91?  oz.     ||     9  |  1  lb.  8oz.  16  pwt.  16  gr.      ||      10  j 

'144.  II  2  cords  2  C.  ft.  4  cu.  ft.  ||  11  |  5iin.  ||  12  |  4  ^  3  3  23  4gr 
144.  II  13  I  lA.  IR.  17P.  21isq.  yd.  ||  14  |  Ipwt.  ISJgr 
1464  1  I  3f  II  2  I  1/^  II  3  I  71  II  4  I  llA  II  5  I  16  ||  6  |  70 
146.  II  7  I  44  II  8  1  1584   |1    9  |  6O8/2   ||    10  |  5987f    ||   11  |  4536 


ANSWKUS.  447 


146.  1  12  I  6405  II  13  |  6975  |  U  |  11725  ||  15  |  3|  |  10  |  12? 
146.  a  n  I  63  II  18  I  1781  ||  19  |  14J-,4  |  20  |  19^  ||  21  |  f  i 
146.  II  22  I  Jj  II  23  I  5'i  II  24  |  ^%  ||  25  j  14  1  26  j  18  |  27  |  130^ 


146.  II  28  I  A  a  29  I  14  ||  30  j  6316^  ||  31  |  j^  \\  32  |  6|| 
146.  II  33  I  I  a  34  I  A  a  35  I  2f  a  36  I  20  a   37  I  fl    II   38  I  jl 

146.  a  89  I  2^V  1  40  I  51  ||  41  |  14^  ||  42  j  If  a   147.  I  43  |  llj 

147.  1  44  I  22|  II  45  |  3,*^  ||  46  j  14f  a  ^^  I  ^  11  ^8  I  »r\ 
147.  II  49  I  55  cents,  a  50  |  34J   a   5'  I  *H  II  52  |  325  a  53  |  i 

147.  a  54  I  ^%  a  55  I  H  a  56  I  H  I  57  |  20^  |  58  |  I2II59  |  5-} 

148.  a  60  I  i  a  61  I  120  A.  =  A's  share  ;  80  A.  =  B's  share  ; 
m.  a  20  A.  =  C'8  share.    ||    149.   ||   1  |  j   a    2  |  A    I    3  j  ,V^ 

149.  a  4  Ufj   1   5  U'A   1    6  I  7^   a    ^  I  36   I    8  I  ^   i    9  I  2^ 

149. a  10 1  iH  a  HI m  ii  12 1 It^  a  13 m  im cit', 

149.  a  15  I  1662^  a  16  I  1383  i  "  ||  a  18  I  M  I  19  I  Hi 
149.  \\  20  I  2if  II  21  I  9tV  II  22  |  48g'^  ||  23  |  ^'^  I  24  |  l^^J- 
149.  a  25  I  rh  II  26  I  tV  II  27  j  ^\  I  28|tV  a  29  j  A  II  30  I  A 
149.  a  31  I  40  II  32  I  1120  ||  33  |  1}  a  34  j  ^  I  35  |  f^^- 
149.  a  36  I  HI-  II  37  I  Hi  a  38  |  Ij  ||  39  |  825^  a  ^0  |  4193tV 
149.  a  41  I  16046^    a    42  I  ?    IM3  I  ^    a    44  I  ^    a    45  I  22f 

149.  a  40  I  68f|f  i  47  j  3^   |1   48  |  1}    j  49  |  9-|   1  50  |  72^';, 

150.  i  51  I  51  lbs,  a  52  I  l^^yds.  a  53  |  1^1  54  |  4  B  55  |  ^ 
150.  a  56  I  3A-  II  57  I  ItV  i  58  I  6  |  59  |  A  ||  60  |  21  a  61  |  27^- 

150.  a  62  I  U^V  a  63  I  ^%  i  64  |  j'y  a  65  |  A  l  151-  11  66  |  ^^ 

151.  a  67  I'lOfa  a  68  I  14j  II  69  |  H  ||  70  |  $H  a  71  I  '3^ 
151.  a  72  I  108/j.  II  73  I  ^  a  74  I  4  a  75  I  24i  i  70  |  l-J  |  77  |  lOJ 


448  ANSWERS. 


151.  II  78   I   41    |n9   I    6    II  80   I    ^    II    81    I   i  II  82   |   6096 

151.  II  83  I  HtV  II  152.  II  1  I  Ij^  I  2  I  Hf  ||  3  |  2|f  ||  4  j  100 

152.  II  5  I  if  II  6  I  f  n  I  Ij  II  8  I  35  II  9  I  ffg-  ||  10  |  2A  |  11  | 

152.  II  531  f  12  I  i,  I  153.  ||  1 1  15  ||  2  |  Hf  ||  3  |  ttf  ||  4  |  42i^ 
53.  II  5  I  ^V^  II  '6  I  26tV  n  I  15    II    8  I  16^  ||  9  |  8bu.  lipk. 

153.  II  10  i  1  mi.  2  fur.  16  rd.  ||  11  |  4mi.  Uur.  19  rd.  3yd.  02Jft. 
153.  II  12  I  20i  II  13  I  14  II  14  |  20Ji  ||  15  |  2700  =  A's  share-, 

153.  II  2800  =  B's  share  ;  800  =  C's  share.    ||    154.    ||    16  |  40 

154.  II  n  I  £n  17s.  5d  Oi-far.  ||  18  |  24  =  John's  ;  32  r=  James' 
154.  II  19  I  285f-  II  20  |  A,  80  ;  B,  24  ;  C,  30  ;  D,  40  ;  66  rem. 
154.  II  21  I  467|  II  22  |  $2j\  sellmg  price  ;  $^%\  =  1st  one's  gain ; 
154.  II  $^^8  =  2d  one's  gain.  ||  23  |  257J|  ||  24  |  7^  ||  25  | 
154.  II  1724i  =  A's  ;  1231f  =  B's  ||  26  |  165  ||  156.  ||  1  |  7ft.  2' 
156.  II  2  I  5  ft.  2'  6"  II  3  |  21  ft.  4'  11"  4'"  ||  4  |  5  ft.  T' 
156.  II  5  I  3' 3"  2'"  II  6  |  2ft.  7' 3"  ||  7  |  15  ft.  4' 10"  4'" 
156.  II  8  I  3ft.  6'  5"  5'"  ||  9  |  87  ft.  10'  7"  4'"  ||  10  |  183ft.  5'  6"  2"' 
156.  II  11  I  223  ft.  8' 4"  9"'  ||  12  |  87  ft.  2'  7"  9"'  6"" 
156.  II  13  I  317  ft.  11' 0"  4'"  II  14  |  543  ft.  6'  3"  2"' sum  ; 
156.  II  |107  ft.  8'  9''  2"'diff.  ||  160.  ||  1  |  41eu.  ft.  3'  10" 
160.  II  2  I  43  sq.  ft.  6'  6"  ||  3  |  82 sq.ft.  9'  4"  ||  4  |  347 sq.  ft.  10'  3" 
160.  1  5  I  554  sq.ft.  7'  8"  8"'  3""  ||  6  |  2917  sq.  ft.  0'  0'^  V'  4."' 
160.  II  7  I  194 sq.ft.  4'  3"  ^"'  \\  8  |  39 sq.ft.  11'  2''  3" 
160.  II  9    I    296  sq.ft.   10'  ^"    \\    10   |   96sq.  yd.  2  sq.ft.  8'  3' 

160.  II  11  I  3150  sq.ft.    ||     12   |   327Jsq.yd.    ||    13    |    21  sq.  ft 

161.  II  14  I  $26.40  II  15  I  10  A.  IR.  25  P.  ||  16  |  3119  sq.ft.  6'  9" 


ANSWERS.  449 


161.  II  11  I  99  II  18  I  $208,011,11  19  |  89cu.  ft.  3'  ||  20  |  I18.49J 

1 , 

161.  II  21    I    504CU.  ft.    II    22    |    11  jj  cords    ||    23    |    24124f? 

161.  II  24  I  41958  ||  25  |  19419cu.  ft.  9'  ||  26  |  849cu.  ft.  8'  8" 

161.  II  27  I  $15,403+  1  28  |  2t5^*^cu.  yd.  ||  162.  ||  29  |    $19,803 

163.  II  1  I  4ft.  7'  II  2  I  5ft.  3'  3"   ||   3  |  48ft.'  6'   ||   4  |  8ft.  7' 

163.  II  5  I  12ft.  6'    II    6  I  37ft.  3'    ||    7   |   1ft.  7'    ||    8   |    8ft. 

163.  II  9  I  6ft.  6'  ^jziW  II  167.  II  1  I  .06  II  2  I  1.7    jj    3  |  .005 

167.  II  4  i  .27  II  5  I  .047  ||  6  |  6.41  ||  7  |  7.008  ||  8  j  9.05  ||  9  1 11.50 

167.  II  10  I  44.7  II  1  I  27.4  ||  2  |  36.015  |1  3  |  99.0027    ||    4  |  .320 

167.  II  5  I  200.000320  ||  6  |  .3600  ||  7  |  5.000003  ||  8  |  40.0000009 

167.  II  9  I  .4900  II  10  I  59.0067    ||    11  |  .0469    ||    12  |  79.000415 

167.  II  13  I  67.0227    ||    14    |    105.0000095    ||    15   |   40.204000 

168.  II  1  I  $37,265  ||  2  |  $17,005  ||  3  |  $215.08  ||  4  |  $275,005 
168.  II  5  I  $9,008  II  6  |  $15,069  ||  7  |  $27,182  ||  8  |  $3,059 
171.  II  1  I  130G.1805  II  2  |  528.697893  ||  3  |  159.37  ||  4  |  1.5415 
171.  II  5  I  446.0924  ||  6  |  27.2087  ||  7  |  88.76257  ||  8  |  71.01 
171.  II  9  I  1835.599  ||  10  |  397.547  ||  11  |  31.02464  ||  12  |  90.210129 
171.  II  13  I  204.0278277    ||    14  |  400.33269960    ||    15  |  .1008879 

171.  II  16  I  $85,463    ||     172.    ||    17  |  $1065.19    ||    18   |    3.8896 

172.  II  19  1  $427,835  ||  20  |  $19,215  ||  21  |  $670,975  ||  22  1  $30,286 

172.  II  23    I    $328,202     ||     24    |    $248,011      ||     25    |    $134,634 

173.  II  1  I  875.0033  ||  2  |  368.5631  !|  3  |  7141.51354  ||  4  1  51.722 
173.  II  5  I  2.7696    ||    6  |   1571.85    ||    7   |   .6946    ||    8   |   .89575 

173.  II  9  I  603.925    ||    10   |   1379.25922    ||    174.    ||    H   |   99.706 

174.  II  12  I  17.949  ||  13  |  .699993    ||    14  |  328.9992    ||    15  |  .999 


450  ANSWERS. 


174.  II  16  I  6314.9  ||  It  |  365,007495  ||  18  |  20.9942 
174.  II  19  I  260.3608?)53  |  20  j  10.030181  ||  21  |  2.0294 
174.  II  22  I  999.999  ||  23  |  2499.75  ||  24  |  103.015  ||  25  |  .4232 
174.  II  26  I  171.925  jj  27  |  $82,625  ||  28  |  $26.60  ||  29  |  126.84194 

174.  I  30   I   $76hl8    II     175.    ||     1   |   .796875    ||    2   |   2.6387  ^ 

175.  11  3  I  .0000500  ||  4  |  1..50050  ||  5  1 26.99178  ||  6  |  10376.283913 

175.  II  7  I  165235.5195  ||  176.  ||  8  |  .0206211250  ||  9  |  28033.797- 

176.  II  099  II  10  I  175.26788356  ||  11  |  .000432045770 
176.  II  12  I  216.94165850  ||  13  |  .000000000294  ||  14  |  18616.74 
176.  II  15  I  933.8253150762  ||  16  |.00715248  ||  17  |  .608785264 
176.  II  18  I  .02860992  ||  19  |  2.435141056  ||  20  |  1296 
176.  D  21  I  312.5  II  22  |  .375  ||  23  |  .0036  ||  24  |  148.28125 
176.  I  25  I   12.13035    ||    26  |  $24.0625    ||    27    j    $3192.005625 

176.  II  28  I  $210.03125  ||  29  |  $708.901875  ||  30  |  $2.06525  gain. 

177.  1  1  I  4796.4  ;  47964  ||  2  |  69472.9  ;  694.729  J  3  |  415300. ; 
177.  1  4153.  II  4  I  2704  ;  27040.  ||  5  |  129072.  ;  1290.72  H  6  | 
177.  II  871000. ;  8710,  |  7  |  140100. ;  1401.  ||  179.  ||  2  |  258.13007 

179.  1  3  I  162.525  ||  4|  2757.89785  U  5  |  3566163  ||  ISO.  ||  1 1  2.22 

180.  I  2  I  8.522  II  3  |  33.331  ||  4  |  1.0001  ||  5  |  12420.5  ||  6  |  .005 
180.  1  7  I  4.25    II    8  I  .007    ||    9  |  .075    ]|    10  j  1.27    ||    11  |  .015 
180.  II  12    I    17.008    U     13    I     25.05068  ;  250.5068  ;  2505.068  ; 
180.  II  25050.68 ;    250506.8      ||      14     |    48.65961  ;    4865.961 
180.  1  48659.61  ;  486596.1  ;  4865961.    ||    15  |  41.622  ;  416.22 


180.  II  4162.2  ;  41622. ;  416220. ;  4162200.  ||  16  |  254.7347748 


180.  II  25473.47748;  254734.7748;  2547347.748;  25473477.48 


ANS\r£K(4.  •  451 


180.  II  2547347U.8    ||     U    |    .1395646+     ||     18   |    1918.515  + 

181.  II  19  I  .004735  ||  20  |  174.412  1  21 1  6^.7125  ||  22  |  1.36832  + 
181.  I  23    I    12976.816+     ||    24    j     .004958+     ||     25    |    6.165 
181.  II  26  I  $9,875  ||  27  |  $2.15   ||   28  |  $.62   ||    29  j  18    I    30  |  8 
181.  1  31  I  14   II  32  I  65.5    ||    33  |  269  acres  ;  $13573.204  cost 
181.  II  $50,458  average  price.  ||  34  |  $7631.8855  share  of  eldest ; 

181.  II  $5723.914125  share  of  others.     ||      182.     [    2    |    10970 

182.  II  3  I  60200  II  4  |  1000    jj    5  |  100    ||    6  |  10  ;  100  ;  1000  ; 
182.  II  30  ;  20  ;  2000  ;  12  ;  1200  ;  500000  ||  183.  ]  3  |  8.311  + 
183.14    I    1.563+       II      5  I  1.16049+       ||      6    j    16.11902+ 
184.  I  1    I     31.69274  ;  3.169274     jj     2    |    57.13562  ;  571.3562 
184.  II  5713.562  ||  3  |  .675  ;  .0675  ;  .0000675  |  4  |  .049  ;  .0049 


184.  11-00049  II  5  I    .030467  ;  .0030467  ;  .00030467  ||  6  |  .004741 
184.  II  .0004741  ;   .00004741      |       7     |    .497  ;    .0497  ;    .00497 

186.  II  1  I  79.1188  II  2  |  35.2843  ||  3  |  11.5834036  1|  4  |  3202.8870 

187.  1  1  I  .25  ;  .5  ;  .75  ||  2  |  .8  ;  .875  ;  .3125  |  3  |  .375  ;  .04 
187.  II  4  I  .015625;  .2666f  ||  6|  .125;  .003  ||  6|  .25714  +  ; 
187-11.44117+  1  7  I  .23903+  ||  8  |  .07157+  ||  9  |  .4375; 
187.  S  .078125  li  10  I  .00448  ||  11  |  .536 ;  .372  ||  12  |  .9 
187.  II  13  I  .73333J  ||  14  |. 48375  ||  15  |  .51282+  ||  16  |  .5375  ; 
187.  S  .005606+  II  17  |  .16666+  J  18  |  1.5555-|  ||  19  |  .15909^, 

187.  II  20  I  $100.80  II  21  |  $17.85  ||  22  |  30.61111  ||  23  |  2.9166? 
187.1  24  I  2.8412+  ||  188.  ||  1  |  j;  f  B  2  |  j ;  |  j  3  ^^  ;  Tii> 
1«8-  II  4  I  ilU  ;  nih  II  5  I  MS-  II  6  I  tI-o    i-  ^  I  tVo^    II    8  I  H 

188.  I  9  I  J  II  10  I  i  II  189.  II  1  I  .0546875  ||  2  |  .325  |  3  |  3.9375 


452  •  ANSWERS. 


189.14  1  .375  il  5  |  71.15113+  ||  6  |.  .(3625  ||  7  |  .15375 
189.  II  8  [3225  II  9'  |  .26175  |1  10  |  100511+  I  11  |  .04 
189.  II  12"|. 91111+  II  13  I  .875  ||  14  1  .01587+  ||  15  |  .712 9 i)  + 
189rj|  16  I  .2325  ||  17  |. 972916+  ||  18  |  .48125  ||  19  |  55 
189.  II  20  I  .001617+  II  21  |  .25625  ||  22  |  .063  ||  23  |  .10416  + 
189.  II  24  1.00994318+  ||  25  |  .791666+  ||  26  |  .3375  I  27  |  .3125 

189.  II  28  I  .040909  ||  29  |  .01875  \\  30  |  .020265  + 
189^  II  31  I  .19672+  ||  32  |  .34895+  ||  33  |  .0153t+  ||  34  |  .005 

190.  Jl  I  2qr.  171b.  4oz.  ||  2  |  Ihhd.  13gal.  3.44  qts. 
190.  II  3  I  16s.  7d.  2.99  far.  ||  4  |  2  gal.  1  qt.  ||  5  | 
190.  II  Iwk.    4  da.    23  hr.    59  min.   56.54+ sec.      ||      6    |    8  P. 


190.  II  7  I  6cwt.  3qr.  ||  8  |  1  lihd.  47  gal.  1  qt.  ||  9  |  20  gal.  1  qt. 

191.  II  10  I  lOoz.  18pwt.  15.99+ gr.  ||  11  |  3qrs.  1.5  iia. 
191.  II  12  I  1yd.  2  ft.  11.9+ in.  ||  18  |  24P.  23sq.  jd.  5sq.  ft. 
191.  II  82.4832  sq.  in.  ||  14  |  32  mi.  1  fur.  14  rd.  4  yd.  2  ft.  9.408  in. 
191.  II  15  I  2ft.  7.5 in.  ||  16  |  43  13  13  9.6+gr. 
191.  1  17  I  3R.  IP.  13.31  sq.  yd.  ||  18  |  9  sheets.  ||  19  |  111b. 
191.  II  20  I  7d.  2ftir.  ||  21  |  1 R.  14  P.  ||  22  |  286da.  Hhr 
191.  II  18min.  36scc.  ||  193.  ||  1  |  .06  ||  2  |  .08125  ||  3  |  .034375 
193.  II  4  I  .01328125  ||  5  |  .0171875  |]  6  |  .034  ||  7  |  .028 
193.  II  8  I  .024219375  ||  194.  1  1  |  .71428571+  ||  2  |  .2666  + 
194.1  3  I  .4545+  ||  4  |  .3888+  ||  196.  ||  3  |  j  ;  /^  ;  jg  ;  jf  ;  ,\ 
^96.  II  4  I   t¥3  ;  t\;  I    II    197.    II    4  I  A;  7  j-g| ;  -jj^;  37fg; 

22  3.     7^4  3  4       II       n      I       3.4    .     217   .      7_    .    412  5^6.    _J6  3_.    AL 
3  3  0?    99399        II       ^      I       45')495>T5>^]G"6  5JI6500>90 


198.  II  2  I    .1875^0'    II     3  |    .0^0344827586206896551724137931' 


ANSWERS.  453 


198.  II  4    I   .'09t56^;..^592';  .5^3'    1    200.    ||    2   |   2.4481818'; 

200.  II  .5^925925';  .008^497133' ||  3  1 165.16^416416';  .04^040404' 
200.11.03^777777'    ||    4    |    .5^333333';  .4^57575';  1.7^5775tr 

201.  II  2  I  95.2^829647'  ||  3 1 69.74^203112'  ||  4  |  55.6^209780437503' 

201.  II  5  I  47.3^763490'  ||  6  j  416.2^542876'  ||  202.  ||  2  |  45.7^755' 

202.  II  3  I  2.9^957'  ||  4  |  1.64ir7'  ||  5  |  .65^370016280907' 
202.  II  6 1  4.37^4'  ||  7  |  4.619^525' ||  8  |  1.0923^'  ||  9  |  1.3462^937' 
202.  II  2  I  5.53780^5'  ||  3  1 1.093^086'"||  4  1 1.64ir7'  ||  5 1 1.7183^39' 
202.  II  6  I    1.4710^037'    ||    7    |   6.r656'    ||    8   |    ll.'^068735402' 

202.  II  9    I    .81654468350'    ||     203.    ||    2    |    13.570413^961038' 

203.  II  3  I  35.024^0'  ||  4  |  7.719^54'  ||  5  |  26.7837^428571' 
203.  II  6  I  3.r45'  II  7  |  3.^8235294117647058'  ||  8  |  1.2^6' 
203.  II  9  I  15.48^423' 


205.  II 
205.  II  " 

21     1           .    II  4|  17      1 
l39~l  +  l~        ,     27~1  +  1~^     , 

205.  II 

l>;-~'                    1  +  1"'       2 

205.  1 
205.  II  ^  1 

47     1       ._             67     1             1  +  1       ^       ^ 
1  65~1  +  1          "2^ '85     1  +  1     ^     g2  +  i~"'^ 

205.  II 

2+l~^     3             3+1-*     ^ 

205.  II 
205.  II 

1  +  1~*_5             H-1     ^  _,, 

1  +  1       ^           „              2  +   1       ^      ,3 

205.  II 

1  +  1-^^      ,,          1  +  1--     , 

205.  II 

1  +  i  ''        i+r" 

205.  II 
205.  II  ^ 
205.  II 

37       1               ,            109     1            , 

1   87       2  +  1      *  "  '  1  450     4  +  1     *      ^ 

2  +  1-*  r  ^           7.+  1     ^    ^ 

205.  II 

1  +  1      «       ,,       1  +  1     ^^    ,, 

205.11                                           5  +  i~^             3  +  l~^'%o 
205.  II  GVf  +  t¥t)  -  2  =  i^SW  Arts.                        1  +  i     '^^ 

454  ANSWERS. 


208.112  i  V  =  4  II  3  I  V  =  5  II  4  \.j\  =  i  ||  5  |  60 -f- 
m.j20Z^  6   I  1^^  =  2    II    7    I    |$|^  =  f gf  ^  Ijl 

mJS    II-  =  f     II     9    I   T^O   =  1     II     10    I   XX  ^    1      II      11    I    30g  ^    2 

208.  II  12  Uf  =  i  l:  13  I  f  II  IM  ^  II  15  I  i  II  16  I  tV  II  H  |  /, 

208.  II  1  I  112  cwt.  II  2  I  5  tons.  ||  209.  ||  3  |  60  ||  4  |  5    ||    5  |  ,? 

209.  II  6  f32  II  7  I  28  II  8  I  $65  ||  211.  ||  1  |  a;  =  60  ||  2  |  ar  =  2T 
209.  II  3  I  ^  =  9  II  4  I  :r  z:::  i  II  5  I  38  II  6  I  56  |i  7  |  12  ||  8  |  40 
213.  II  1  I  or  =  21  II  215.  II  1  I  330  II  2  I  90  II  3  I  504  ||  4  |  2.08 

215.  II  5  I  875  II  216.  ||  6  |  99  ||  7  |  27621  ||  8  |  20  ||  9  |  122.85 

216.  II  10  I  1400  11  11  I  164«5  ||  12  |  121.875  ||  13  |  $27 
216.  II  14  I  710Z.  II  15  I  3533.936  ||  16  |  86.62  ||  17  j  £39679  10s. 

216.  II  18  I  9||19|8Jrd.||  20  |  160  yds.  ||  21  |  7-1  ||  217-  ||  22  |  10 

217.  II  23  I  920  II  24  |  54  ||  25  |  39.375  ||  26  |  382.85  ||  27  |  63 
217.  II  28  I  $.036  II  29  |  $7080.48  [  30  |  $1,925  ||  31  |  2.10 
217.  II  32  I  52.50    ||    33  |  $f|     ||    34  |  7200    ||    35  |  $37,909  + 

217.  II  36  I  225  ||  37  |  20   ||  218.   ||   38  |  54    ||    39  |  12   1|  40  |  6 

218.  II  41  1 160  II  42  I  40.47  ||  43  |  10  yr.  ||  44  |  51  ||  45  1 132.589-f 
218.  II  46  I  1121  II  47  |  18.66|  ||  48  |  66.355  ||  49  |  106f  ||  50  [  40 

218.  II  51  I  112.86   II   52  |  18090    ||    219.    ||    53  |  21  gal.    ||    54  I 

219.  II  2142  to  A  ;  1125  to  B  ||  55  |  .625  ||  56  |  6f  ||  57  |  $15.86| 
219.  II  58  I   168  lbs.    ||    59   |   93f    ||    60    |  552    ||    61    |    17444 

219.  II  62  I  6hr.  32mm.  43j-\sec.  ||  63  |  140  ||  220.  ||  64  |  l^da 

220.  II  65  I  221  da.  ||  66  |  45  ||  67  |  Uj\  \\  68  |  20i-  ||  69  |  10^ 
220.  II  70  I  131  II  71  I  810  II  72  I  6  II  73  |  lOlir.  40  min.  Se^^^sec 
220.  II  74  I  16^  times.  ||  222.   ||   1  |  I6I  ||   2  |  7200   ||   3  |  1871 


ANSWERS.  455 


t 


222.  II  4  I  72  II  5  I  10  i  6  I  92J  II  223.  t  |  36  |  8  |  292.5  ||  9  |  156 

223.  II  10  I  9600  II  11  I  50  ||   12  |  13f  ||   13  |  8571f  J  14  |  3  hr. 

223.  1  15  I  $471.04  ||  16  |  3if  ||  224.  ||   17  |  180  J  18  |  13^  in. 

224.  I  19  I  14  j  II  20  I  7^  ||  21  |  97i  ||  22  |  32  j  23  |  32  ||  24  1 132 
226.  II  2  I  $1000,  A's;  81200,  B's  ;  $800,  C's  ||  3  1 1714.28f,  A's 
226.  II  285.7 If,  B's  ||  4  |  $4030,  A's  ;  $3980,  B's  j  $3980,  C's  , 
226.  II  $4010,  D's    II    5    |    100,  A's  ;  140,  B's  ;  200,  C's    ||    6  | 

226.  II  $3333i     1st  ;     $3000,  2d  ;     $3000,    3d  ;     $2666|,    4th 

227.  II  7  I  $3000,  widow's  ;  $1500,  son's  |  8  |  $12961.50,  A's; 
227.  II  $15737.25,  B's ;'  $10802.25,  C's  ;  $1833,  D's  gain. 
227.  II  9  I  $450,  A's  ;  $600,  B's ;  $750,  C's  ||  10  |  4242.50,  A  s 
227.  II  stock  ;  1697,  A's  gain:  5939.50,  B's  stock;  2375.80  B's 
227.  II  gain  :  6788,  C's  stock  ;  2715.20,  C's  gain.  ||  11  |  237.75, 
227.  II  A's  ;  181.0625,  B's  ;  125.4375,  C's  ;  70,  D's.  jj  12~j 
227.  II  87.831+    A's;      65.06+    B's;      48.795,    C's;      68.313, 

227.  II  D's.      II      13    I    2553,  A's  ;    3401.70,  B's  ;    1405.30,  C's. 

228.  II  14  I  15063f,  B's  ;  9586J,  A's.  ||  15  |  1015.331  the  first  ; 
228.  11  1523.00,  the  second  ;  2030.66f ,  the  third.  ||  2  |  16.38,  A's  ; 

228.  II  35.10,  B's;  18.72,  C's.  ||  229.  ||  3  |  $7  ||  4  |  6577.23fJ5, 

229.  1  A's  ;  1822.76111,  B's.  ||  5  |  288,  A's  ;  270,  B's  ;  240,  C's. 
229.  II  6  I  280,  D's  ;  168,  C's.  ||  7  |  2648.86^^,  A's  ;  2901.13/^. 
229.  II  B's  ;  1850,  C's.  |  8  |  $800,  B's  stock  ;  15mo.,  C's  time. 
231.  I  1  I  50.24  II  2  I  114.78  ||  3  |  1.1875  ||  4  |  2.839375 
231.  II  5  I  1.002  II  6  I  12  II  7  I  90  1  8  I  16.74  ||  9  |  47.725 
231.  I  10  I  27.54    II    11    I    300.365    ||    12   |   15.75    ||    13  |  160 


456  ANSWERS. 


231.  II  14  I  478.125  ||  15  |  4344.35  ||  16  |  2625  ||  11  |  5144.625 
231.  II  18  I  12500  II  19  |  3867.018t5  ||  20  |  15000  ||   21  |  22.95 

231.  II  22  I  43.20  ||  23  |  65  ||  24  |  U2.85  ||  25  |  205 
i^l  I  20  II   2  I  121   II   3  I  H  IM  I  13f  II   5  I  25  II   6  1  87i 

232.  II  T  I  571  II  8  I  f  "A  ||  9  [  47/g  ||  10  |  331-  ||  11  j  3ti  ||  12  |  12^ 

233.  II  13  I  8tJ  II  14  I  80  |1  15  |  70  ||  16  j  66f  |1  It  |  160 
233.  II  2  I  1900    II    3  1100    [j  4  |  400  ||  5   |    15000   ||   6  |  142f 

233.  II  t  I  1.9^  II  8  I  90    II    9  I  1800    ||    10  |  4392    ||    11  |  20800 

234.  II  1  I  388.1188  ||  2  |  9000  ||  3  |  550  ||  4  |  156  ||  5  |  30.123-f 

235.  II  6  I  140  II  1  I  3ni1.77J  ||   8  |  5400   ||  9  |  4  ||   10  |  5425 

235.  II  1  I  160  II  236.  i|  2  |  150  |1  3  |  950  \\  4  |  30000  \\  5  |  13500 

236.  II  6  I  50000   ||    1  |  5000    ||    8  |  2600    ||    237.    ||    1  |  33.15 

237.  II  2  I  21.411  II  3  |  236.25  ||  4  j  6.15/^  ||  5  |  $.695  per  bushel 

238.  II  6  I  8166  ||  1  |  $.56  ||  8  j  915.15  ||  9  |  133.20 
238.  II  10  I  11165.311  II  11  I  444.15  ||  12  |  1910.115  ||  13  |  2cts. 

238.  II  14  I  18860  ||  15  j  11%  ||  16  |  90  cents.  ||  239.  ||  11  |  $3.20 

239.  II  18  I  $.96  II  19  |  $18.03  ||  20  |  $.66  ||  21  |  $1.80  ||  22  |  18% 
239.  II  23  I  25%     ||    24    |    Neither.    ||    25  |  8'0%    ||    26  |  25  % 

239.  II  21  I  160.34315  gain  ;  4f  %    |1    240.    ||    28  |  $1041.15908i 

240.  II  29  I  25%  on  gold;  20%  on  paper.  ||  30  |  1612.90ii^ 
240.  II  31  I  14980  ||  32  |  10562.50  ||  33  |  20000    ||    34  |  260000 

240.  II  35    I    11%    II    36  I  $426    ||     31    j    $400    ||    38    |    110 

241.  II  39  I  548.80  ||  40  |  350,  1st;  525,  2d  ;  10,  gain.  ||  41  |  6 
241.  11  42  I  4315  ||  43  |  45%  ||  44  |  25.65  lost.  ||  45  |  3450,  cost; 
241.  II  5%  loss.  I  46  I  40  %  ||  41  |  lOf  %  ||  48  j  339,  cost ;  508.50 


ANgWERS.  457 


242.  II  1  I  188.50,  com.  ;    ^1351.50  paid  over.  ||  2  |  40.n,  com.  ; 

242.  1  1359  laid  out.      jj      3    j    34.8375      jj      4    |    164.53125 

243.  II  5  I  96.33;  5831.67  ||  6  |  163.80,  com.  ;  4340.70,  cost. 
243.11  7  I  115.391    ||    8  |  6835.283    ||    9  |  935    ||    10  |  420.922 

243.  II  11  I  $70  II  12  I  2571.36  ||  13  |  39.1875,  charges  ;  1267.062£ 

• — — — 1 

243.  II  trans.  |i  244.  ||  14  |  11764.705+  ;  235.295+  ||  15  |  15  tons 

244.  II  16  I  63.625,  com. ;  4544.642+  bu.  J  17  |  158bbls.  , 
244.  II  $2412.66  ||  18  |  183.0607+ ;  3.6612+  ||  19|2^8_t''/» 
244.  II  20  I  2-^YU  \\  21. 1  8f  %  ||  22  |  llJVo  ||  246.  ||  1  |  43.875 
246.  II  2  I  60.9875  ||  3  |  224.91  ||  4  |  360.2832  ||  5  |  473.844 
246.  II  6  I  1312.50  ||  7  |  283.8438  ||  8  |  422.8976  ||  9  |  1112.90 
246.  II  10  I  265.2345  ||  11  |  1893.75  ||  12  |  373.2495  ||  13  |  735 
246.  II  14  I  1016.075  ||  15  |  120.80  ||  16  |  5796  ||  17  |  20.909 
246.  II  18  I  26.313  ||  19  |  458.88  ||  20  |  1979.5013J  ||  21  |  5618.75 
246.  1  22  I  628.416J  ||  23  |  64.0625  ||  24  |  157.65625 
249.  II  2  I  42.24325  ||  3  |  420.253125  ||  4  j  213  ||  5  |  181.25 
249.  II  6  I  11.0415  II  7  |  132.7707+  ||  8  |  26.9586+  ||  9  | 
$49.  II  416.1673+  ||  10  |  |334.2187+  ||  11  |  120.0693  + 
249.  II  12  I  40.0968  ||  13  |  81.'6778+  ||  14  |  162  ||  15  |  221.266 
249.  B  16    I    389.2406     ||     17  |  135.3714      \\      18    |    42.9404  + 

249.  II  19  I  84.6855  \\  20  |  55.6685+  j  21  |  32.666? 
2.50.  II  22   I  8590.8333J    ||    23  |  36    ||    24  |  93.7843+    ||    25 

250.  II  160.4408+  ||  26  |  12.963+  ||  27  |  82.036+  |  28  |  70.964 
250.  II  29  I  879.46703+  ||  30  |  801.769  ||  31  |  933.1573  + 
250.  II  32    I    499.339+     ||     33    |    140.6444+     ||     34    |    5085 


458  ANSWEKS. 

251.  II  35  I  403.858  \\  36  |  9337.50  ||  1  |  394.325625  ||  2  |  697.986 

251.  II  3  I   3339.613  ||  252.  |i  4   |   823.902+  ||  5   |  4640.532.-{- 

252.  II  6 1 1976.6305+  |i  2|ie45  8s.  Ifcl.  1|  253.  ||  3  |  ^£45  12s.  4d.+ 

253.  (I  4  I  iei54  7s.  Od.  2far.  ||  5  |  £1133  10s.  9id.  ||  6| 
253.  II  £199  6s.  3fd.  ||  7  |  £6  16s.  5d.  j|  255.  ||  2  ]  5359.3664  + 

255.  II  3  I  8925.5443:  ||  4  |  1127.041   ||  5  |  190.758  \\  256.  ||  6  | 

256.  II  156.20+  II  257.  |1  2  |  3976.782|  ||  3  |  439.80  ||  4  |  6234.76 

257.  II  5  I  30000  ||  6  |  952.576+  ||  7  |  7%  ||  8  |  10%  ||  9  |  51  % 

258.  II  10    I    121%    II     11    I    2yr.  6mo.     |]     12    |    16yr.  8mo. 

258.  II  13  I   5yr.  4mo.    ||    14   |    1  yr.  6'mo.  20  da.    ||    15   |    7500 

259.  II  2  I  25.3575  ||  3  |  291.7215  1|  4  ]  57.3048  [  5  |  73.0154  + 
259.  II  6  I  83.20  ||  7  j  $845.8376+  ||  8  |  $48165.9388+  ||  9  | 
259.  II  $14523.55509+  ||  260.  ||  1  |  562.50  ||  2  |  184.499  + 
261.  II  3  I  21  II  4  I  5000  ||  5  |  1902.587+   ||  6  |  236.438  =  dis..; 


261.  II  2763.562  pres.  value.  ||  7  j  1379.6123+  ||  8  |  3538.0835  + 
261.  II  cash  value  ;  388.0835+  gain.  ||  9  1  9890.23864  + 
261.  II  10  I  .00414+  at7J  cts.  ||    11  |  13.33J    ||    12  |  2369.2617, 


261.  II  cash  value  ;  61.9883,  diff.  ||  265.  ||  1  [  6.15  ||  2  |  7.65 
265.  II  3  I  23.2913  dis.;  476.708  pres.  value,  jj  4  |  1225.3555 
265.  II  5  I  4.375  |i  6  |  82.5916  gain.   ||    7  |  11.785    ||    8  |  15.4044 

265.  il  9  I  981.21  ||  10  ]  474.375  ||  266.  ||  2  |  296.50  ||  3  |  697.20 

266.  II  4  I  1041. 666f  ||  5  |  3522.092  ||  268.  ||  2  |  387.  ||  3  |  90 
268.  11  4  1  2559.06,  A's  ;  3210.6375,  B's.    ||    5  |  153    i|    2  |  5320 

268.  il  3  I  666  II  4  1  17455.50   ||   269.    ||    5  |  59110    ||    6  j  21375 

269.  II  7  I  7999.6875  ||  8  |  213500   |1    9  |  307    ||    2  |  3529.411  + 


ANSWERS.  459 


270.13    I    5G    II    4   I    4000    ||    5    j   7235.142+     ||     6   |   8000 

270.  II  7  I  10432.432+    ||    2  |  8»/«    ||    271.    ||    3  18V.   ||   4  |  8"/. 

271.  II  5  15%  II  272.  ||  2  |  20%  ||  3  |  41 J  %  ||  4  |  12i%  premium. 

272.  II  2  I  7 "/»  best.  1   3  |  8%  best.,  ||   4  |  166.66f   ||  274.  ||    1 
•274.  II  5168.59  ||  2  |  158.40  ;  237.60  ||  3  |  252  ;  126   ||   4  |  300 

274.  1  5  I  89.55  ||  6  |  47.811  ||  7  |  1252.12^  ||  8  |  163.80 
274-  II  9  I  16481.25  ||  10  |  5^%  ||  275.  ||    11  |  l|°/o  ||  12  |  44% 

275.  II  13  I  24000  f  14  |  9020  ||  15  |  127.4625  ||  16  |  298.2546 
277.  II  1  I  121.72  II  2  |  232.50  ||  3  |  262.50  ||  4  |  20  ||  5  |  98.20 
277-1  6  I  120  II  7  I  9101.635  ||   278.  ||  1  \  411.15  ||  2  |  757.908 

279.  II  3  I  1227.395  ||   280.  ||    1  |  7051.63415    ]   2  |  9049.53795 

280.  II  3  I  23058.6765  ||  281.  ||  4  |  2195.95  ||  5  |  2159.613  + 
i82.  II  1  t  H  %  II  2  I  37901125  ||  283. 1  3  |  1^% ;  82.25,  A's  tax  ; 
283.  156.9075,  B's  tax.  j  4  |  f  %  ;  15.50  ||  6  |  5820  i  6  | 
283.  II  22236.197  ||  7  |  4656.05,  whole  tax  ;  5  mills  on  $1  ;  $27  ; 

283.  1  6.8775,  G's  tax  ;  12.78,  H's  tax.  1  284.  ||  8  m  cts.  on  $1  ; 

284.  II  112.50  ;  18.  ||  9  |  7.40  ;  9.225  ||  285.  ||  1  |  12  mo.  ||   2  |  9 

285.  I  3  I  8|mo.  ||  4  |  7mo.  3da.  ||  5  |  6Jmo.  ||  6  |  6mo.  6da. 

286.  II  7  1  26  J  da.,  or  July  28th.    ||   287.   ||   2   |   28^^.  or  ^pril 

287.  II  29th,  Eq.  time  of  purchase  ;  Dec.  29th,  Eq.  time  of  paym*t, 

287.  II  3  I  78^ da.,  or  Oct.  18th.  ||  288.  1  4  |  76yYTV/H  ^a.,  or 

288.  II  July  13th.  ||  5  |  21^1^,  or  21  days.  ||  2  |  9-Jmo. 
2«9.  II  3  I  5^  mo.  ]    2  |  25  mo.  J    3  |  5f  mo.  |   4  |  421  da.,  from 

289.  II  Jan.  1st ;  or  on  Feb.  26d,  next  year.  ||  5  |  July  8th,  1857. 
291.  II  2  I  $.58  hit.  balance  ;  $700.58  cash  balance.  1|  292.   1   3  | 


460  ANSWERS. 


292.  II  '!>46.20  int.  balance  ;  1403.80  cash  balance.   ||    4  j  $.10  int. 

292.  II  balance  ;  $620.70  cash  balance.   ||   293.   ||    1  |  109  da.  from 

293.  II  April  2d,  or  Dec.  14, 1860. ||    294.    ||    2  |  2449.15  balance  ; 

294.  II  April  9th.  ||  3  |  July  13th.  ||  295.  ||  1  |  8.50  ||  2  |  $  06 
296.  II  3  I  $.49  II  4  I  $1.00  ||  5  |  75°  ||  6  |  19  ||  7|^^.ir,i 
296   II  8  I  $.30  loss.  ||    298.   ||   1  j  1  lb.  at  8  cts. ;    1  lb.  at  lOcts. ; 

298.  II  31b.  at  14  cts.    ||    2  |  1  lb.  of  each.    ||    299.    ||    3  |  1  calf, 

299.  II  2  cows,  1  ox,  1  colt.  ||  4  |  3  gallons.  ||  300.  ||   1  |  20  lb.  of 

300.  II  each  ||  2  |  75  lb.  of  each.  ||  3  |  3G  gal.  at  7s. ;  24  gal.  at 
300.  II  7s.  Cd.  and  9s.  Gd. ;   12  gal.  at  9s.  ||  4  |  10  at  $2,  and  15  at 

300.  II  $f  II  5  I  25  lb.  at  5  and  7  ;  100  lb.  at  1^  •  37^  at  9J  ;  and 
SCO.  II  50  at  10    I    301.    ||    1  |  221b.  of  each.   ||   2  |  9  gal.  water, 

301.  II  401  gal.  at  $2.50,  and  IS^gal.  at  $3.00  ||  3  |  12  sheep,  16 
301.  II  lambs,  12  calves.  ||  4  |  8  at  $6,  8  at  17,  4  at  $19 
301.  II  5  I  90  gal.  at  4s.,  and  10  gal.  each  at  6s.,  8s.,  and  10s. 
301.  II  6  I  6  vests,  12  pants,  6  coats.  ||  7  |  30  at  15,  4  each  of  20, 
301.  II  22,  and  24  ||  8  |  10  at  $^,  15  at  $1,  10  at  $5 
304.  II  3  I  1260.9932  ||  4  |  713.37  ||  5  |  6T.  14cwt.  1  qr.  16.681b. 

304.  II  6    I    6T.  13cwt.  2qr.  4  1b.  ;  $308.4774      ||      7    |   792.612 

305.  II  8  I  1196.343+    ||    9  |  255.835+    ||     10  |  4.09+    ||    11] 

305.  II  398.1199  ||  12  |  466.27875  ||  13  1 1101.24  gain ;  14  cts.,  price 

306.  II  14  I  7936.50  i|  15  |  820.4625  ||  16  |  423.36  ||  17  |  251.453-+ 

306.  II  18   I   1457.75    ||     19   |    22.605  cwt.,  tare  ;  $68.5856  dutv 

307.  II  1  I  225 j^  tons.  ||  2  |  4384|  tons.  ||  3  |  729gV5  tons. 
307.  II  4  I  1006.57+  tons.  ||  316.  ||  1  |  8591.975  ||  2  |  8637.16875 


ANSWERS.  461 


316.  II  3  I  9177.036  ||  4  1  9970  ||  5  |  $15000.305   ||  6  |  9801.9299-f 

317.  11  2    I    176204.4729+     ||    318.    ||    3    |    jeUOU  17s  7id. 

318.  II  4     I     0005.368+     ||    5   |   807.874+    ||    6   |   9096.806  + 

319.  II  2  I  7^'o  premium.  ||  3  |  12280.06  ||  4  |  84597  francs  66 
319.  I  centimes.  ||  1  j  6057.693  ||  2  |  1250.52  ;  3%  nearly  belovf 
319.  II  par.    ||    321.    ||    2  |  5761.31  +  florins.    ||    3  |  9962.219+ 

322.  II  4    I    3495.839+  Sp.  doll.    ||    323.    [     1  |  16    |    2  |  225 

323.  II  3  I  20164  ||  4  |  214369  ||  5  11795000  |  6  |  605.16 
323.  II  7  I  .276676  ]  8  |  9.765626  ||  9  j  .00274576  ||  10  |  j% 
323.  I  11  I  H  II  12  U?  II  13  I  igf  I  II  14  I  mU  II  15  |  58.140625 
323.  II  16  I  250^2^^  II  17  |  51030.81  ||  18  |  216  ||  19  |  13824 
323.  II  20  I  1953125  ||  21  |  2515456  ||  22  |  20736  ||  23  |  59049 
323.  II  24  I  76.765625  ||  25  |  10.4976  ||  26  |  .0184528125 
323.  II  27  I  iUi  II  28  |  j^-  \\  29  |  ^U^  ||  30  |  57^^, 
323.  II  31  I  II^Ht  II  32  |  14880.930  ||  33  |  .000244140025 
323.  II  34  I  2893640.625  ||  327.  ||  3  |  7  ||  4  |  12  ||  5  |  15  ||  6  |  48 
327.  II  7  I  89.409+  ||  8  |  2505  ||  9  |  137.84+  ||  10  |  1003.8677  + 
327.  II  11  I  191.713+  II  12  I  1000  ||  13  |  311.011+  ||  14  |  173.853  + 
329.  II  4  i  f  II  5  I  if  II  6  I  .14  ||  7  |  2.5  ||  8  |  16.7  ||  9  |  .453  ||  10| 
329.  II  .93+  II  11  1.9682+  ||  12|.1581+  ||  13  |  •>/5l  =  2J.  Arts, 


329.  II  14|  ^."7994  =  . 89409 +^ns.  II  lb\YJ22l  =  A1U+  Aus. 
329.  II  16  I  .779+    ||    17  |  .149+    ||     18   |  5.01    |    19   |    14.015 
329.  II  20  I  1.2247+   ||   21  |  fj   ||   22  |  f  ||  23  |  .2828+   ||    24 
329.  II  11.618+     II    25    |    .885+    ||    26    |- 75.15    ||    27   |   400.06 
331.  1  1  I  343  II  2  I  221  j  3  I  21  ^P  II  4  I  00  rd.  wide  ;  180  rd.  long. 


462  ANSWERS. 


331.  II  5  I  40  rows  ;  80  trees  in  a  row  ;  10  A.  0  R.  29  P.  168f  sq. 
33?.  II  ft.  area.  ||  6  |  75  ft.    ||    332.    ||     7  |   135    ||    8  j  94.708  + 

332.  II  9  I  53.33^- ft.  ||  XO  |  8.66+  ft.  ||  11  |  825.8  mi.  ||  12  |  ilOO 
332^1113  175    11    14  I  28.28+ ft.    ||    333.    ||    15  |  6 in.    ||    16  | 

333.  i  11.041+ rd.  ||  17  |  4.405+ in.,  1st  man's  share: 
333.  1  5.739+  in.,  2d  man's  share  ;  13.856+ in.,  3d  man's  share. 
336.  II  1  I  12  II   2  I  49   II   3  I  36   II   4  I  247   ||   5  |  179   ||   6  |  364 

336.  I  7  I  439  II  8  I  3072'  }  337.  ||   1  1  2.028+   I   2  |  12.0016+ 

337.  1  3  I  .232+  1  4  |  27.0002+  ||  5  |  .729+  ||  6  |  .015 
337.  II  7  I  .188+  II  8  |  4.339+  ||  1  j  j  ||  2  J  |-  i|  3  |  3|||  4  [Tj 

337.  II  5  I  I  II  6  I  ^^  II  7  I  fi  II  8  |  jj  II  9  I  1  987+  ||  10  |  3.83  + 

338.  i  1  I  27  ft.  II  2  I  19  ft.  long  ;  2166  sq.  ft.  area.  ||  3  |  36  ft., 
338.  i  leng-th  of  side.  ||  4  |  8.57+ it.  ||  5  I  9.77+ ft.,  length  ; 
338.  II  19.54+  ft.,  height.  ||  6  |  10.125  cu.  ft.  ||  7  |  45cts.  per  yd. ; 
338.  II  2025  yd.  j  9  |  641b.   ||    10  |  8  ft.,  length  of  side.   ||    11  |  8 

338.  II  12    I    $1331    II    13   |    12  in.  long;  6  in.  wide;  1  in.  thick. 

339.  II  14  I  24  ft.  long  ;  20  ft.  wide  ;  9  ft.  deep.  i|  15  |  20  ft. 
339.  II  16  I  .54+ in.,  1st  woman's  share;  .69+ in.,  2d  woman's 
339.  II  share;  .99+ in.,  3d  woman's  share;  3.77+ in.,  4th 
339.  II  woman's  share.  ||  340.  ||  1  |  89  ||  2  |  $80  ||  341.  ||  3  |  $396 

341.  1  4  I  $1550  II  5  I  171  rd.  ||  6  |  201ft.  ||  342.  ||   1  |  5  miles. 

342.  II  2  I  $2    II    3  I  I  inches.    ||     4   |   15,  18,  21,  24,  27,  30,  S3 

343.  II  1  I  2730  II  2  |  226  last  term  ;  $64.96  whole.  ||  3  |  791imi 

343.  II  4  I  10  mi.  7  fur.  27  rd.  1|  yd.      ||      344.      i|      1    |    5551 

344.  i  2  I  13  da.  ;  312  m.    i|  3  |  6  ||  346.  ||   1  |  1  ||  2  |  3125000 


ANSWERS.  4(53 


346.  S  3  I  ^V   I  4  I  $100000000000    ||    5  |  $3200    ||    6  |  $54000 

346.  1  1  I  8327.68  ||  8  |  $595,508  1|  347-  ||  1  |  118081  1  2  1  2044 

347.  B  3  I  11184810   II  4  |  $42949072.95    ||    348.    ||    5  |  938249- 


348.  i  9224-  ships.  ||  355.  ||  16  |  $4.4954  ||   359.  ||  22  |  4166.40 

359.  I  23   I   9f    II    24  |  36  ft.    |    25  |  1770    ||    360.   ||    26  |  2if 

360.  j  27  1  4.33^  |  28  |  100  |  29  j  5500  yd.  ||  30  |  117  ||  31  |^ 
360.  1  32  I  Ucts.  II  33  |  $56  ||  34  |  $5600  J  35  |  7^  cts.  ||  36  | 

360.  II  843  I  37  I  $246.75  gain.  ||  361.  |1  38  1  5hr.  27min.  16i\sec. 

361.  11  39  I  40  II  40  I  10  II  41  I  36  days.  ||  42  |  l-J-f 
361.  II  43  I  60  =  1st  part ;  100  =  2d  part  ;  140  =  3d  part  ; 
361.  I  180  =  4th  part.  ||  44  j  16|  in.  ||  45  |  25\rao.  ||  46  |  $2.20 

361.  J  47  I  $12  II  48  |  $1.20    ||    362.    ||    49  |  78.652 -f-  discount. 

362.  1  50  I  $129.60  J  51  |  $19,375  most  advantageous  on  time» 
362.  1  52  I  292.823  gain  ||  53  |  122.70  =  A's  share  ;  163.00  - 
362.  I  B's  share  ;  196.32  =  C's  share.  ||  64  j  $2317.15  =  A's  ; 
362.  II  $1853.72  =  B's  ;  $2317.15  =  C's  ;  $2780.58  =  D's. 
362.'  II  55     I    $95.10  =  A's  ;     $95.10  =  B's  ;     $133.14  =  C's  ; 

362.  B  152.16  =  D's.  J  56  |  7ioz.  1  57  |  8|  da.    ||    58  |  17  times. 

363.  I  59  I  4}f  da.  1  60  |  5  mo.  24  da.  ||  61  |  68  da.  J  02  |  126 gal. 

363.  i  63  I  $172.78  loss  on  stocks.  J  64  |  $3312.417+  1  65  |  $42.60 

364.  I  66  I  4  yd.  ||  67  |  140  miles.  |  68  |  $2,  the  1st ;  $6,  the 
364.  J  second.  J  69  |  100  ||   70  ]  $3825  ||  71  |  $144  gain  by  bor- 

364.  II  rowing.  J  72  |  $36000  ||  73  |  41.183-f   B   36.5.  |   74  |  Tiie 

365.  I  second,  10  days  after  the  3d  ;  this  1st,  8  days  after  the 
365.  J  2d,  or  18  days  after  the  3d.   j]    75  |  $6890  ||  76  |  100  A., 


484 


ANSWE-HS. 


1st  Co.  ;   88A.,  2dCo. ;   $7  per  acre.    ||    11   \   Uda.,   or 


March  16th.  ||  IS  |  512  slabs  ;  $302.22|  |1  19  \  350,  A's  ; 


210,  C's  ;  291.50,  B's  ;   175,  D's  ;  122.50,  E's.   ||    80  |  72 


81  I  5hr.  20min.  p.m.    ||    82  |  S0.66f  |  83  |  24  chickens 


and  36  turkeys.  ||  366.  ||  84  |  8  days.  |]  85  |  1797.50,  1st, 


2157,  2d  ;  2516.50,  3d.  ||  86  |  |640  stock,  and  $120  gain 


2d.  ;  $960  stock,  and  $180  gain,  1st.   ||    87  |  49.945+  ft. 


f  wk.  II  89  I  llf  hr.  ;  1341-  miles.    ||    90  |  533^-,  A's  ; 


888f,  B^s  ;  177J,  C's.    ||    91  |  36^  days. 


367. 


92 


84485.006+  ft.    ||    93  |  $206.06+  in  favor  of  1st  inyest. 


94     I     23599680  cu.  yd.      ||      95    |    4646.363  + 


96 


555.017+,  1st;  4354.717+,  2d;  4304.663  +  ,  3d; 


5781.263+,  4th;  4004.338  +  ,  5th. 


97     I    2160 


98  I  $564,  A's  ;  $423,  B's.  ||  368.  ||  99  |  $31  ||  100  |  8hr. 


101  I  $30  com.  diff.  ;  $246f  cost.     ||     102    |    $14461.50^ 


103  I   97Jlb.    II    104  I  f  ct.,cost;  ^ct.  sold  for;    -^oCt., 


gain  on  each  ;  80  eggs  sold. 


105    I     84  years  old. 


106  I  942.48+  cubic  feet. 


107  I  155A.  311.  38.72  P. 


108  I  $365,837+    ||     109  |  6/g-bours.    ||     110  |  5  inches 


111  I  $4006.54+  II  371.  i  2|36A.||3|5A.  IK  15P.  ||  4 


371. 

II  135  A.  II 372.  II 

1|437A. 

2R.  34.32+P.  II  2 1 291A.  2R.  16P 

372. 

B3 

35  A.  OR. 

25  P.    II 

4  1  20A.    11   5  1  40A.    || 

6  1  15A. 

372. 

n 

24A.  IR. 

8  P.      II 

8     1     26  A.  3R.  20  P. 

5  sq.  yd. 

ANSWERS.  465 


372.  II  9  i  120  feet.     ||     373..  ||     2    |    21  A.  OR.  39.824P.  ||  3  j 

373.  II  921.875 sq.ft.  ||  4  |  704.125 sq.  yd.  ||  5  |  GOA.  3R.  12.8P. 

373.  II  6  I  270  A.  IR.  24  P.  ||  374.  ||  2  |  584.3376  ||  3  |  125.664 

374.  il  4  I  179.0712    ||     1  |  50    ||    2  |  7418    ||    3   |    4300.8354- 
71.  II  1  I  113.0976  II    2  I  19.035  ||  3  j  153.9384    ||    375.  J  4   j 

;75:  II  1.069+     II     5    I    20  A.  OR.  16.9984 P.     ||     1    |    113.076 

375.  II  2  I  615.7536  11  3  |  4071.5136  ||  4  |  196996571.722104  sq.  mi. 

376.  II  2  I  268.0832     ||     3  |  2144.6656  cu.  in.    ||    4  |  259992792- 

376.  II  082.6374908  ||  5  |  .9047808  cu.  ft.  ||  377.  ||  1  |  9100  sq.  ft. 

377.  II  2  I  1440  sq.  ft.     ||     2|    110592  cu.  in.     ||     3    |    42f  cu.  ft. 

377.  II  4  I  315f|gal.    ||    5  |  13820cu.  ft.    ||    378.    ||    1  |  2513.28 

378.  II  2  I  233.33Jsq.  ft.  ||  3  |  2827.44  sq.  in.   ||  4  |  6283.2  sq.  ft. 

379.  II  2  I   36442.56      ||      3    |    13571.712      ||      4    j    9650.9952 

379.  II  5    I    7363.125     ||     2    |    4380    ||    3  |  2484     ||    4    |    5620 

380.  II  5  I  5760  ||  6  |  14400  ||  7  |  1800  ||  2  |  9160.9056 
380.  II  3  I  8659.035  ||  4  |  2827.44    ||    382.    ||    2  |  32.4938  inches. 

382.  II  3  I  28.2574  in.    ||    383.    ||    1  |  197.459+ gal.  wine.    ||     2| 

383.  II  136.9209+  gallons  wine;  112.7583+  gallons  beer. 
383.  II  3  I  148.3772+ gal.  wine.  ||  385.  jj  1  |  401b.  ||  2  |  251b. 
38.5.  II  3  I  501b.  II  4  j  201b.  ||  5  |  401b.  ||  6  |  lin.;  l^in.;  2 in.;  4 in. 
386.  II  7  I  641b.  ||  8  |  1501b.  ||  388.  |  1  |  60  lb.  ||  2  |  401b 
388.11  3  I  251b.    II    1   I   7ift.    ||    389.    ||    2  |  IJft.    ||    1    |   40  Ih. 

389.  II  2  I  1001b.    II    3  |  601b.    ||    390.    ||    1  j  576    ]    2   |   2250 

390.  II  3  I  lOGGf  II  4  |  3000  ||  391.  ||  1  |  2592001b.  ||  2  |  1.47+  lb. 
392.  1  3  I  1.15+ lb.  II  4  |  1.2  in.  ||    393.   ||    1  |  3G9fi  ft. ;  2316ft. 


466  ANSVVEUS. 


394. 

Il-^i 

3G18fft. 

;  482ift.   II   3  |223f|lft.i| 

4  1  2^  sec.  nearly. 

394. 

II  H 

1608i-ft. 

space  ;   32 If  velocity.  ||   6 

1  2  mi.  4984i|3ft. 

394. 

ni 

164.69  J II 

8  1  100.52iLft.    II    9  1  397151  ft;  4{||  sec. 

:?94. 

II  10 

1  160|  ft. 

=  velocity  ;    402^V¥fi  ft- 

II     11    1    197J-^n 

m. 

II  12 

1  77;-f|see.    II     13   1    1904i2^«3  ft- = 

height;    101511  :: 

394. 

II  time  of  ascent.      ||      14    |    94jV3sec. 

II       15    1    1447.5 

395 

II  16 

1  61.24  sec.    II     17  1  14.28+ sec.    || 

18  1    15050-ii|rt. 

395. 

II  19 

1  12  sec; 

2316  ft.  II  396.  1  r|  8.857 

II   2|38l|fcu.  fi. 

397. 

II  3  1 

.980  II  4  1 

2  ft.  11.388  in.  II  5  |  190  T 

'.  709  11).  II  ()  i  2.75 

397.  II  7  1  7.234  + 

II    8  1  .786+11    9  1  .875     || 

10  1  177  11).  5(»z 

397. 

II  11 

1  1.103  II 

398.   i   1  1  3.49  qt.    ||    2  | 

37.5  II  3  1  2.4r)  oT 

398. 

1  4  1  JA  =  .5319. 

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TERMS  OF  EXAMlNATiOK. — We  propose  to  supply  any  teacher  who  deBircs  to 
examine  tcvt-booka,  with  a  vinw  to  introduciion,  if  a2>2)rvvcd,  with  sample  copies,  on 
receipt  of  onk-ii.vlf  the  ])rico  annexed  (in  Catalogue),  and  Iho  books  will  be  sent  by 
mail  or  express  without  expense  to  the  purchaser.  Books  marked  thus  (*)  are  ex- 
cepted from  this  offer. 

TERMS  OF  INTRODUCTION.— The  Publishers  are  prepared  to  make  special  and 
vory  favorable  terms  for  first  introduction  of  any  of  llie  Natioxal  Seuies,  and  will 
furnish  the  reduced  introductory  price-list  to  teachers  whose  application  presents  evi- 
dence of  good  faith. 

Teachers  desiring  to  avail  themselves  of  any  of  the  prirrilefjcif  of  the  prof ession^  il 
not  known  to  the  Publishers,  should  mention  the  name  of  one  or  more  of  their  Trus- 
tees or  Patrons,  as  pledge  of  good  faith. 

For  farther  information,  address  the  ruhlishers. 


The  JVatio7ial  Se^-ies  of  Standard  Sc?iool-^ooks, 

ORTHOGRAPHY  AND  READING. 


JSTATIOFAL  SERIES 

OP 

READERS    AND     SPELLERS, 

BY  PAEKEE  &  WATSOK 


The  National  Primer $25 

National  First  Reader 38 

National  Second  Reader 63 

National  Third  Reader 95 

National  Fourth  Reader 1  50 

National  Fifth  Reader 1  88 

National  Elementary  Speller 25 

National  Pronouncing  Speller 45 


Tliis  unrivaled  series  has  acquired  for  itself  during  a  very  few  years  of  publication, 
a  reputation  and  circulation  never  before  attained  by  a  series  of  school  readers  in  the 
Bame  space  of  time.  No  contemporary  books  can  be  at  all  compared  -with  them.  Tho 
average  annual  increase,  in  circulation  exceeds  100,000  volumes.  We  cliallengo  rival 
publishers  to  show  such  a  record. 

The  salient  features  of  these  works  which  have  combined  to  render  them  so  popular 
may  be  briefly  i-ecapitulated  as  follows  ; 

1.  THE  WORD  METHOD  SYSTEM— Tins  famons  progressive  method  for  young 
children  originatad  and  was  copyrighted  with  these  books.  It  constitutes  a  process  by 
which  the  beginner  with  toords  of  one  letter  is  gradually  introduced  to  additional  lists 
formed  by  prefixing  or  affixing  single  letters,  and  is  thus  led  almost  insensibly  to  tho 
mastery  of  tlio  more  difficult  constructions.  This  is  justly  regarded  as  one  of  tluj 
most  striking  modern  improvements  in  methods  of  teaching. 

2.  TREATMENT  OF  PEONUKOIATION.— The  wants  of  the  youngest  scholars 
in  this  department  are  not  overlooked.  It  may  be  said  that  from  the  first  lesson  the 
student  by  this  method  need  never  be  at  a  loss  for  a  prompt  and  accurate  render- 
ing of  every  word  encountered. 

3.  ARTICULATION  AND  ORTEO^T  are  recognized  as  of  primary  in*. 
Jtortanca  ^t  *      ^ 

3  (0»«>C) 


TAe  JVatioiiut  Series  of  Standard  Sc?ioot-2)Ooks, 

ORTHOGRAPHY   AND   READING-Continued. 

4.  PUNCTUATION  is  inculcated  by  a  scries  of  interesting  reading  lessong.  the 
simple  perusal  of  which  suffices  to  fix  its  principles  indelibly  upon  tke  mind. 

5.  ELOCUTION.  Each  of  the  higher  Readers  (3d,  4th  and  5th)  contains  elaborate, 
Echolarly,  and  thoroughly  practical  treatises  on  elocution.  This  feature  alone  has 
secured  for  the  series  many  of  its  warmest  friends. 

6.  THE  SELECTIONS  are  the  crowning  glory  of  the  Eeries.  Without  exception 
it  may  be  said  that  no  volumes  of  the  same  size  and  character  contain  a  collection  bo 
diversified,  judicious,  and  artistic  as  tliis.  It  embraces  the  choicest  gems  of  English 
literature,  so  arranged  as  to  afford  the  reader  ample  exercise  in  every  department  of 
style.  So  acceptable  has  the  taste  of  the  authors  in  this  department  proved,  not  only 
to  the  educational  public  but  to  tlio  reading  community  at  large,  that  thousands  of 
copies  of  the  Fourth  and  Fifth  Headers  have  found  their  way  into  public  and  private 
libraries  throughout  the  country,  where  they  are  in  constant  use  as  manuals  of  liter- 
ature, for  reference  as  well  as  perusal. 

7.  ARRANGEMENT.  Tlic  exercises  arc  so  arranged  as  to  present  constantly  al- 
ternating practice  in  the  different  styles  of  composition,  while  observing  a  definite 
plan  of  progression  or  gradation  throughout  the  whole.  In  the  higher  books  the  ar- 
ticles are  placed  in  formal  sections  and  classified  topically,  thus  concentrating  the  in- 
terest and  inculcating  a  principle  of  association  likely  to  prove  valuable  in  subsequent 
general  reading. 

8.  NOTES  AND  BIOGRAPHICAL  SKETCHES.  Tliese  are  full  and  adequate 
to  every  want.  The  biograpliical  sketches  present  in  pleasing  style  the  history  of 
every  autlior  laid  under  contribution. 

9.  ILLUSTRATIONS.  These  are  plentiful,  almost  profuse,  and  of  the  highest 
character  of  art  They  are  found  in  every  volume  of  the  scries  as  far  as  and  including 
the  Third  Reader. 

10.  THE  GRADATION  is  perfect  E.ich  volume  overlaps  its  companion  pre- 
ceding or  following  in  tlie  series,  so  that  the  scholar,  in  passing  from  one  to  another, 
is  barely  conscious,  save  by  the  presence  of  the  new  book,  of  the  transition. 

11.  THE  PRICE  is  reasonable.  The  books  wore  not  trimmed  to  the  minimum 
of  size  in  order  tliat  the  publishers  might  be  able  to  denoniinate  them  "  the  cheapest 
in  the  market,"  but  were  made  large  enoxcgh  to  cover  and  suffice  for  the  grade  indi- 
cated by  the  respective  numbers.  Thus  the  child  is  not  compelled  to  go  over  his  First 
Reader  twice,  or  be  driven  into  the  Second  before  he  is  prepared  for  it  The  compe- 
tent teachers  who  compiled  the  series  made  each  volume  just  what  it  should  be,  leav- 
ing it  for  their  brethren  who  should  use  the  books  to  decide  what  constitutes  true 
cheapness.  A  glance  over  the  books  will  satisfy  any  one  that  the  same  amount  of 
matter  is  nowhen.  furnished  at  a  price  more  reasonable.  Besides  which  another  con- 
sideration enters  into  the  question  of  relative  economy,  namely,  the 

12.  BINDING.  By  the  use  of  a  material  and  process  known  only  to  themselves, 
in  common  with  .ill  the  publications  of  this  house,  the  National  Readers  are  warranted 
to  out-last  any  with  which  they  may  be  compared—the  ratio  of  relative  durability  be- 
ing in  their  favor  as  two  to  one.  ^ 


The  JVatto7iat  Seizes  of  Standm'd  Sc?ioot-!Books, 

SCHOOL-KOOM    CARDS, 

To  Accompany  the  National  headers. 
^-•-♦^••^- 

Eureka  Alphabet  Tablet *i  50 

Presents  tha  alphabet  upon  tlie  Word  Method  System,  by  which  the 
child  will  learn  the  alphabet  in  nine  days,  and  make  no  small  progress  in 
reading  ana  spelling  in  the  same  time. 

National  School  Tablets,  lo  Nos *7  60 

Embrace  reading  and  conversation;'.!  exercises,  object  and  moral  les- 
sons, form,  color,  &c.  A  complete  set  of  these  large  and  elegantly  illus- 
trated Oards  will  embellish  the  school-room  more  than  any  other  article 
of  furniture. 


READING 


Fowle's  Bible  Reader $l  00 

The  narrative  portions  of  the  Bible,,  chronologically  and  topically  ar- 
ranged, judiciously  combined  with  selections  from  the  Psalms,  Proverbs, 
and  other  portions  which  inculcate  important  moral  lessons  or  the  great 
truths  of  Christianity.  The  embarrassment  and  difficulty  of  reading  the 
Piide  itself,  by  course,  as  a  class  exercise,  are  obviated,  and  its  use  made 
feasible,  by  this  means. 

North  Carolina  First  Reader 50 

North  Carolina  Second  Reader 75 

North  Carolina  Third  Reader i  oo 

Prepared  expressly  for  the  schools  of  this  State,  by  C.  11.  Wiley,  Super- 
intendent of  Common  Schools,  and  F.  M.  Hubbard,  Professor  of  Litera- 
ature  in  the  State  University. 

Parker's  Rhetorical  Reader i  oo 

Designed  to  familiarize  Readers  with  the  pauses  and  other  marks  in 
general  use,  and  lead  them  to  the  practice  of  modulation  and  inflectien  of 

the  voice. 

Introductory  Lessons  in  Reading  and  Elo- 
cution      75 

Of  similar  character  to  the  foregoing,  for  less  advanced  classes. 

School  Literature l  50 


High 


Admirable  selections  from  a  long  list  of  the  world's  best  writers,  for  ex- 
ercise in  reading,  oratory,  and  composition.  Speeches,  dialogues,  and 
model  letters  represent  the  latter  department. 

5 


90 


77ie  JVational  Series  of  Sta7idard  Schoot-^ooks, 

ORTHOGRAPHY. 

SMITH'S    SERIES 

BuppHcs  a  Bpeller  for  every  class  in  graded  schools,  and  comprises  the  most  coin> 
"  pleto  and   excellent  treatise    on    English   Orthography  and  its  companion 
branches  extant. 

t.  Smith's  Little  Speller $20 

Finfit  Itonnd  in  the  Ladder  of  Learning. 

2.  Smith's  Juvenile  Definer 45 

Lessons  composed  of  familiar  words  grouped  with  reference  to  similar 
Bi<rnific;itif)n  ol*  use,  and  correctly  spelled,  accented,  and  defined. 

3.  Smith's  Grammar-SchooI  Speller ....     so 

Familiar  words,  grouped  with  reference  to  the  samenoss  of  e-^nnd  of  syl- 
lables difftirently  spelled.  Also  definitions,  complete  rules  for  spelling  and 
formation  of  dirivatives,  and  exercises  in  false  orthograjjhy. 

4.  Smith's  Speller  and  Definer's  Manual    • 

A  complete  School  DictioiKtry  containing  14,000  words,  with  various 
other  useful  matter  in  the  way  of  Kules  and  Exercises. 

5.  Smith's  Hand-Book  of  Etymology  .    .      i  25 

The  first  and  only  Fitymology  to  recognize  the  Anglo-Stizon  onr  mother 
toiigne;  containing  also  full  lists  of  derivatives  from  the  Latin,  Greek, 
Gaelic,  Swedish,  Norman,  &c.,  &c. ;  being,  in  fact,  a  complete  etymology 
of  the  language  for  schools. 

Sherwood's  Writing  Speller 15 

Sherwood's  Speller  and  Definer is 

Sherwood's  Speller  and  Pronouncer    ...      15 

Tlio  Writing  Speller  consists  of  properly  ruled  and  numbered  blanks 
to  receive  the  words  dictated  by  the  tejicher,  with  space  for  remarks  and 
corrections.  The  other  volumes  may  be  used  for  the  dictation  or  ordinary 
class  exercises. 

Price's  English  Speller *15 

A  complete  spelling-book  for  all  grades,  containing  more  matter  than 
"Webster,"  manufactured  in  superior  style,  and  sold  at  a  lower  price — 
consequently  the  cheapest  speller  extant. 

Northend's  Dictation  Exercises ^^ 

Embracing  valuable  information  on  a  thousand  topics,  communicated 
In  such  a  manner  as  at  once  to  relieve  the  exercise  ©f  spelling  of  its  usnal 
tedium,  and  combine  it  with  instruction  of  a  general  character  calculated 
to  profit  and  amuse. 

Wright's  Analytical  Orthography    ....     25 

Tills  standard  work  is  popular,  because  it  teaches  the  elementary  soiindB 
in  a  phiin  and  philosophical  maimer,  and  presents  orthography  and  or- 
thoepy in  an  easy,  uniform  system  of  analysis  or  parsing. 

Fowle's  False  Orthography  .......     45 

Exercises  for  correction. 

Page's  Normal  Chart- *3  75 

The  elementary  sounds  of  the  language  for  the  school-room  walls. 

6 


2'he  jVatlonal  Series  of  Standai^d  School-!Books* 


ENGLISH  GRAMMAR. 


CLAKK'S  DIAGRAM   SYSTEM. 

Clark's  First  Lessons  in  Grammar    ...     50 

Clark's  English  Grammar i  oo 

Clark's  Key  to  English  Grammar  ....  60 
Clark's  Analysis  of  the  English  Language  •  60 
Clark's  Grammatical  Chart 4  oj 

The  theory  and  practice  of  tc^aching  grammar  in  American  schools  is 
meeting  with  a  thorough  revolution  from  the  use  of  this  system.  While 
the  old  methods  offer  proficiency  to  the  pupil  only  after  much  weary 
plodding  and  dull  memorizing,  tliis  affords  from  the  inception  the  ad- 
vantage of  practicdl  Object  Teaching,  addressing  the  eye  by  means  of  il- 
lustrative figures  ;  furnishes  association  to  the  memory,  its  most  power- 
ful aid,  and  diverts  the  pupil  by  taxing  his  ingenuity.  Teachers  who  are 
using  Clark's  Grammar  uniformly  testify  that  they  and  their  pupils  find 
it  the  most  interesting  study  of  the  school  course. 

Like  all  great  and  radical  improvements,  the  system  naturally  met  at 
first  with  much  unreasonable  opposition.  It  has  not  only  outlived  the 
greater  part  of  this  opposition,  but  finds  many  of  its  warmest  admirers 
among  those  who  could  not  at  first  tolerate  so  radical  an  innovation.  All 
it  wants  is  an  impartial  trial,  to  convince  the  most  skeptical  of  its  merit. 
No  one  who  has  fairly  and  intelligently  tested  it  in  the  school-room  has 
ever  been  known  to  go  back  to  the  old  method.  A  great  success  is  al- 
ready established,  and  it  is  easy  to  prophecy  that  the  day  is  not  far  dis- 
tant when  it  will  be  the  only  system  of  teaching  English  Orammar.  As 
the  System  is  copyriglited,  no  other  text-books  can  appropriate  this  ob- 
vious and  great  improvement. 

Welch's  Analysis  of  the  English  Sentence  •  i  lo 

llcniarkable  for  its  new  and  simple  classification,  its  method  of  treat- 
ing connectives,  its  explanations  of  the  idioms  and  constructive  laws  of 
the  language,  &c. 

ETYMOLOGY. 


Smith's  Complete  Etymology, i  25 

Containing  the  Anglo-Saxon,  French,  Dutch,  Gorman,  'Welsli,  Danish, 
Gothic,  Swedish,  Gaijlic,  Italian,  Latin,  and  Greek  Roots,  and  the  English 
words  derived  therefrom  accurately  spelled,  accented,  and  defined. 

The  Topical  Lexicon, i  50 

This  work  is  a  School  Dictionary,  an  Etymology,  a  compilation  of  syn- 
onyms, and  a  manual  of  general  information.  It  differs  from  the  ordinary 
lexicon  in  being  arranged  by  topics  instead  of  the  letters  of  the  alphabet, 
thus  realizing  the  apparent  paradox  of  a  "  Readable  Dictionary."  An 
unusually  valuable  school-book. 

7 


The  JVatio7ial  Seines  of  Standai'd  School-Sookt, 


GEOGRAPHY 


TUE 

NATIONAL  GEOGRAPHICAL  SYSTEM. 


I.  Monleilh's  First  Lessons  in  Geography,  %    35 

II.  Monteith's  Introduction  to  the  Manual,  .     65 

III.  Monteith's  New  Manual  of  Geography,  .  i  oo 

IV.  Monteith's  Physical  &  Intermediate  Geog.  i  75 
Y.  McNally's  System  of  Geography,    ...  i  88 

The  only  complete  course  of  geographical  instruction.  Its  circnlation 
Is  almost  universal — its  merits  patent.  A  few  of  the  elements  of  its  popu- 
larity are  found  iu  the  following  points  of  excellence. 


1.  PRACTICAL  OBJECT  TEACHING.  The  infant  scholar  is  first  introduced 
to  a  picture  whence  he  may  derive  notions  of  the  shape  of  the  earth,  the  phenomena 
of  day  and  night,  the  distrihution  of  land  and  water,  and  the  great  natural  divisions, 
which  mere  words  would  fail  entirely  to  convey  to  the  untutored  mind.  Other  pic- 
tures follow  on  the  same  plan,  and  the  child's  mind  is  called  upon  to  grasp  no  ide» 
without  tho  aid  of  a  pictorial  illustration.  Carried  on  to  tho  higher  books,  this  system 
culminates  in  No.  4,  where  such  matters  as  climates,  ocean  currents,  the  winds,  pecu- 
liarities of  tho  earth's  crust,  clouds  and  riin,  are  pictorially  explained  and  rendered 
apparent  to  the  most  obtuse.  The  iIlustra|lons  used  for  this  purpose  belong  to  the 
highest  grade  of  art. 

2.  CLEAE,  BEAUTIFUL,  AND  CORRECT  MAPS,  in  the  lower  numbers 
the  maps  avoid  unnecessary  detail,  while  respectively  progressive,  and  aCfording  the 
pupil  new  matter  for  acquisition  each  time  he  approaches  in  the  constantly  enlarging 
circle  the  point  of  coincidence  with  previous  lessons  in  the  more  elementary  books. 
In  No.  4,  the  maps  embrace  many  new  and  striking  features.  One  of  the  most 
effective  of  these  Is  tho  new  plan  for  displaying  on  each  map  tho  relative  sizes  of 
countries  not  represented,  thus  obviating  much  confusion  which  has  arisen  from  the 
necessity  of  presenting  maps  in  the  same  atlas  drawn  on  different  scales.  The  maps 
of  No.  6  have  long  been  celebrated  for  their  superior  beauty  and  completeness.  This 
is  the  only  school-book  in  which  the  attempt  to  make  a  complete  atlas  also  clear  and 
distinct,  has  been  successful.  The  map  coloring  throughout  the  series  is  also  notice- 
able. Delicate  and  subdued  tints  take  the  place  of  the  startling  glare  of  inharmonious 
colors  which  too  frequently  in  such  treatises  dazzle  the  eyes,  distract  the  attention, 
and  serve  to  overwhelm  the  names  of  towns  and  the  natural  featurea  of  the  landscape. 

8 


2'he  J^atlonal  Series  of  Standard  SchoolSooks, 


GEOGRAPHY-Continued 

3.  THE  VAKIETY  OF  MAP  EXERCISE.  Starting  each  time  from  a  differenl 
basis,  the  pupil  in  many  instances  approaches  the  same  fact  no  less  than  six  times, 
thus  indelibly  impressing  it  upon  his  memory.  At  the  same  time  this  system  is  not 
allowed  to  become  wearisome — the  extent  of  exercise  on  each  subject  being  graduated 
by  its  relative  importance  or  difficulty  of  acquisition. 

4.  THE  OHAEACTEE  AND  ARRANGEMENT  OF  THE  DESCRIPTIVE 
TEXT.  The  cream  of  the  science  has  been  carefully  culled,  unimportant  matter  re- 
jected, elaboration  aroided,  and  a  brief  and  concise  manner  of  presentation  cultivated. 
Tlie  orderly  consideration  of  topics  has  contributed  greatly  to  simplicity.  Due  atten 
tion  is  paid  to  the  facts  in  history  and  astronomy  which  are  inseparably  connected 
with,  and  important  to  the  proper  understanding  of  geography — and  such  only  are 
admitted  on  any  terms.  In  a  word,  tho  National  System  teaches  geography  as  a 
science,  pure,  simple,  and  exhaustive. 

5.  ALWAYS  UP  TO  THE  TIMES.  The  authors  of  these  books,  editoriaUy 
speaking,  never  sleep.  No  change  occurs  in  the  boundaries  of  countries,  or  of  coun- 
ties, no  new  discovery  is  made,  or  railroad  built,  that  is  not  at  once  noted  and  re- 
corded, and  the  next  edition  of  each  volume  carries  to  every  school-room  the  new  or- 
der of  things. 

6.  SUPERIOR  GRADATION.  This  is  the  only  scries  which  furnishes  an  avaU- 
able  volume  for  every  possible  class  in  graded  schools.  It  is  not  contemplated  that  a 
pupil  must  necessarily  go  through  every  volume  in  succession  to  attain  proficiency. 
On  tlie  contrary,  two  will  suffice,  hut  tliree  are  advised  ;  and  if  the  course  will  admit, 
the  whole  scries  should  be  pursued.  At  all  events,  the  books  are  at  band  for  selection, 
and  every  teacher,  of  every  grade,  can  find  among  them  one  ezactbj  suited  to  his  class. 
The  best  combination  for  those  who  wish  to  abridge  the  course  consists  of  Nos.  1,  3, 
and  6,  or  where  children  are  somewhat  advanced  in  other  studies  when  they  com- 
mence geography,  Nos.  2,  3,  and  5.  Where  but  two  books  are  admissible,  Nos.  2  and 
4,  or  Nos.  3  and  5,  are  recommended. 

7.  FORM  OF  THE  VOLUMES  AND  MECHANICAL  EXECUTION.  The 
maps  and  text  are  no  longer  unnaturally  divorced  in  accordance  with  the  time-hon- 
ored practice  of  making  text-books  on  this  subject  as  inconvenient  and  expensive  as 
possible.  On  the  contrary,  all  map  queslioi>s  are  to  be  found  on  the  page  opposite  the 
map  itself,  and  each  book  is  complete  in  one  volume.  The  mechanical  execution  is 
unrivalled.  Paper  and  printing  are  everything  that  could  be  desired,  and  the  bind- 
kig  is— A.  S.  Barnes  and  Company's. 


Ripley's  Map  Drawing $1  25 

This  system  adopts  the  circle  as  its  basis,  abandoning  the  processes  by 
triangulation,  the  square,  parallels,  and  meridians,  &c.,  which  have  been 
proved  not  feasible  or  natural  in  the  development  of  this  science.  Suc- 
cess seems  to  indicate  that  the  circle  '•  has  it." 

National  Outline  Maps 

Foi  thfi  school-room  walls.    In  preparation. 

9 


The  National  Seines  of  Standard  Schooi-^ooks* 

MATHEMATICS. 

AKITHMETIC. 

1.  Davics'  Primary   Arithmetic .    .    .$    2o 

2.  Davies'  Intellectual  Arithmetic 40 

3.  Davies'  Elements  of  Written  Arithmetic 50 

4.  Davies'  Practical  Arithmetic 1  00 

Key  to  Practical  Arithmetic *1  00 

5.  Davies'  University  Arithmetic 1  50 

Key  to  University  Arithmetic *1  50 

ALGEBRA. 

1.  Davies'  New  Elementary  Algebra 1  26 

Key  to  Elementary  Algebra *1  25 

2.  Davies'  University  Algebra 1  60 

Key  to  University  Algebra *1  GO 

3.  Davies'   Bourdon's    Algebra 2  25 

Key  to  Bourdon's  Algebra *2  25 

GEOMETRY. 

1.  Davies'  Elementary  Geometry  and  Trigoncrtietry   .  1  40 

2.  Davies'  Legendre's   Geometry 2  25 

3.  Davies'  Analytical    Geometry  and  Calculus     ....  2  50 

4.  Davies'  Descriptive   Geometry 2  75 

MENSURATION. 

1.  Davies'  Practical  Mathematics  and  Mensuration    .    .    .  1  40 

2.  Davies'  Surveying    and    Navigation 2  50 

3.  Davies'  Shades,   Shadows,  and   Perspective  .    .         .    .  3  75 

MATHEMATICAL    SCIENCE. 

Davies'   Grammar  of  Arithmetic *    50 

Davies'    Outlines  of  Mathematical  Science *1  00 

Davies'    Logic  and    Utility  of  Mathematics *1  50 

Davies  &  Peck's  Dictionary  of  Mathematics *3  75 

10 


7%e  JVa^onat  Series  of  Sta7idard  Sc?iool'-^ooks, 

DAYIES'  NATIONAL  COUESE  of  MATHEMATICS. 

ITS     RECORD. 

In  claiming  for  this  series  the  first  place  among  American  text-books,  of  whatever 
class,  the  Publishers  appeal  lo  the  magnificent  record  which  its  volumes  have  earned 
during  the  thirty-five  years  of  Dr.  Charles  Davies'  mathematical  labors.  The  unre- 
mitting exertions  of  a  life-time  have  placed  the  modern  series  on  the  same  proud  emi- 
nence among  competitors  that  each  of  its  predecessors  has  successively  enjoyed  in  a 
course  of  constantly  improved  editions,  now  rounded  to  their  perfect  fruition — for  it 
Bccms  indeed  that  this  science  is  susceptible  of  no  further  demonstration. 

During  the  period  alluded  to,  many  authors  and  editors  in  this  department  have 
started  into  public  notice,  and  by  borrowing  ideas  and  processes  original  with  Dr. 
Davies,  have  enjoyed  a  brief  popularity,  but  are  now  almost  unknown.  Many  of  tho 
series  of  to-day,  built  upon  a  similar  basis,  and  described  as  "modem  books,"  are 
destined  to  a  similar  fate  ;  while  the  most  far-seeing  eye  will  find  it  difficult  to  fix  tho 
time,  on  the  basis  of  any  data  afforded  by  their  past  liistory,  when  these  books  will 
cease  to  increase  and  prosper,  and  fix  a  still  firmer  hold  oa  the  affection  of  every 
educated  American. 

One  cause  of  this  unparalleled  popularity  is  found  in  the  fact  that  the  enterprise  of 
the  author  did  not  cease  with  the  original  completion  of  his  books.  Always  a  practi- 
cal teacher,  he  has  incorporated  in  his  text-books  from  time  to  time  tho  advantages 
of  every  improvement  in  methods  of  teaching,  and  every  advance  in  scioncc.  During 
nil  the  years  in  which  he  has  been  laboring,  he  constantly  submitted  his  own  theorica 
and  those  of  others  to  the  practical  test  of  the  class-room — approving,  rejecting,  or 
modifying  them  as  the  experience  thus  obtained  might  suggest.  In  this  way  he  has 
been  able  to  produce  an  almost  perfect  scries  of  class-books,  in  which  every  depart- 
ment of  mathematics  has  received  minute  and  exhaustive  attention. 

Nor  has  he  yet  retired  from  the  field.  Still  in  the  prime  of  life,  and  enjoying  a  rlpo 
experience  which  no  other  living  mathematician  or  teacher  can  emulate,  his  pen  is 
ever  ready  to  carry  on  the  good  work,  as  the  progress  of  science  may  demand.  AVit- 
ness  his  recent  exposition  of  the  "  Jletric  System,"  wTiich  received  the  ofiicial  cu- 
dorsemeat  of  Congress,  by  its  Committee  on  Uniform  Weights  and  Measures. 

Davies'  System  is  tiii!  ackxowledqed  National  Standaeo  fo3  the  United 
States,  for  the  following  reasons : — 

1st.  It  is  the  basis  of  instruction  in  the  great  national  Gchools  at  West  Point  aad 
Annapolis. 

2d.  It  has  received  tho  quasi  endorsement  of  the  National  Congress. 

8d.  It  is  exclusively  used  in  the  public  schools  of  the  National  Capital. 

4th.  The  officials  of  the  Government  use  it  as  authority  in  all  cases  involving  mathe- 
matical questions. 

5th.  Our  great  soldiers  and  sailors  commanding  the  national  armies  and  navies 
were  educated  in  this  system.  So  have  been  a  majority  of  eminent  scientists  in  this 
country.    All  these  refer  to  "  Davies"  as  authority. 

6th.  A  larger  number  of  American  citizens  have  received  their  education  from  thia 
than  from  any  other  series. 

7th.  The  series  has  a  larger  circulation  throughout  the  wholo  country  than  any 
othei",  being  extensively  imd  in  every  State  in  the  Union, 


The  JVatio7ial  Series  of  Standat'd  SchoolSooks, 

MATHEMATICS-ContinuBd. 

ARITHMETICAL    EXAMPLES. 

Reuck's  Examples  in  Denominate  Numbers  %    50 
Reuck's  Examples  in  Arithmetic ....      l  00 

These  volumes  differ  from  the  ordinary  arithractie  in  their  peculiarly 
practical  character.  They  are  composed  mainly  of  examples,  and  afford 
the  most  severe  and  thorough  discipline  for  the  mind.  While  a  book 
which  should  contain  a  complete  treatise  of  theory  and  practice  would  ba 
too  cumbersome  for  every-day  use,  the  insuftlcicucy  of  j»aciicaZ  examples 
has  been  a  source  of  complaint 

HIGHER     MATHEMATICS. 

Church's  Elements  of  Calculus 2  50 

Church's  Analytical  Geometry 2  so 

Church's  Descriptive  Geometry,  with  Shades, 

Shadows,  and  Perspective 4  50 

These  volumes  constitute  the  "  West  Point  Course"  in  their  several 
departments. 

Courtenay's  Elements  of  Calculus    ....  3  25 

A  work  especially  popular  at  the  South. 

Hackley's  Trigonometry  3  oo 

With  applications  to  navigation  and  surveying,  nautical  and  practical 
geometry  and  geodesy,  and  logarithmic,  trigonometrical,  and  nautical 
tables. 

THE     METRIC     SYSTEM. 

The  International  System  of  Uniform  Weights  and  Measures  must  hereafter  be 
taught  in  all  common-schools.  Professor  Charles  Davies  is  the  official  exponent  of 
the  system,  as  indiaated  by  the  following  resolutions,  adopted  by  the  Committee  of  the 
House  of  Representatives,  ona  "  Uniform  System  of  Coinage,  Weights,  and  Measures^" 
February  2,  18G7  :— 

Resolved,  That  this  committee  has  observed  with  gratification  the  efforts  made  by 
the  editors  and  publishers  of  several  mathematical  works,  designed  for  the  use  of  com- 
mon-schools and  other  institutions  of  learning,  to  introduce  the  Metric  System  of 
VVeights  and  Measures,  as  authorized  by  Congress,  into  the  system  of  instruction  of 
the  youth  of  the  United  States,  in  its  various  departments ;  and,  in  order  to  extend 
further  the  knowledge  of  its  advantages,  alike  in  public  education  an*  in  general  use 
by  the  people, 

Le  it  further  resolved,  That  Professor  Charles  Davies,  LL.D.,  of  the  State  of  New 
York,  be  requested  to  confer  with  superintendents  of  public  instruction,  and  teachers 
of  schools,  and  others  interested  in  a  reform  of  the  present  incongruous  system,  and, 
by  lectures  and  addresses,  to  promote  its  general  introduction  and  use. 

The  official  version  of  the  Metric  System,  as  prepared  by  Dr.  Davies,  may  be  found 
in  the  Written,  Practical,  and  University  Arithmetics  of  the  Mathematical  Scries,  and 
ta  also  published  separately,  price  postpaid,  ylre  cent3. 

12 


The  JVational  Se?^les  o/  Standa7*d  SchoolSooks, 


HI  ST  OB  Y. 

Monteith's  Youth's  History, $75 

A  History  of  the  United  States  for  begiuners.  It  is  arranged  upon  the 
catcchetic.il  plan,  with  illustrative  maps  and  engravings,  review  questions, 
dates  in  parentheses  (tliat  their  study  may  be  optional  with  the  younger 
class  of  learners),  and  interesting  liiographical  Sketches  of  all  persons 
who  have  been  prominently  identified  with  the  history  of  our  country, 

Willard's  United  States,  School  edition,  ...  i  25 

Do.  do.  University  edition,      .  2  25 

The  plan  of  tliis  standard  work  is  chronologically  exhibited  in  front  of 
the  title-page ;  the  Maps  and  Sketches  are  found  useful  assistants  to  the 
memory,  and  dates,  usually  so  difficult  to  remember,  are  bo  systematically 
arranged  as  in  a  great  degree  to  ol)viate  the  dilhculty.  Candor,  impai'- 
tiality,  and  accuracy,  arc  the  distinguishing  features  of  the  narrative 
portion. 

Willard's  Universal  History, 2  25 

The  most  valuable  features  of  the  "  United  States"  arc  reproduced  in 
this.  The  peculiarities  of  \\\z  work  arc  its  great  conciseness  and  the 
prominence  given  to  the  chronological  order  of  events.  The  margin 
marks  each  successive  era  with  great  distinctness,  so  that  the  pupil  re- 
tains not  only  the  event  but  its  time,  and  thus  fixes  the  order  of  history 
firmly  and  usefully  in  his  mind.  Mrs.  AVi Hard's  books  are  constantly 
revised,  and  at  all  times  written  up  to  embrace  important  historical 
events  of  recent  date. 

Berard's  History  of  England, i  ^^ 

By  an  aulhuress  well  known  for  the  success  of  her  TTistory  of  the  United 
States.  The  social  life  of  the  English  people  is  felicitously  intei-woven, 
as  iu  fact,  with  the  civil  and  military  transactions  of  the  realm. 

Ricord's  History  of  Rome, i  25 

Possesses  the  charm  of  an  attractive  romance.  The  Tables  with  which 
this  history  abounds  are  introduced  in  such  a  way  as  not  to  deceive  the 
inexperieaced,  while  adding  materially  to  the  value  of  the  work  as  a  reli- 
able index  to  the  character  and  institutions,  as  well  as  the  history  of  the 
Soman  people. 

Banna's  Bible  History, l  25 

The  only  compendium  of  Bible  narrative  which  affords  a  connected  and 
chronological  view  of  the  important  events  there  recorded,  divested  of  all 
superfluous  detail. 

Alison's  History  of  Europe 2  50 

An  abridgment  f  )r  Schools,  by  Gould,  of  this  great  standard  work, 
covering  the  eventful  period  from  A.  U.  17S9  to  18i5,  being  mainly  a  his- 
tory of  the  career  of  Napoleon. 

Marsh's  Ecclesiastical  History, i  gb 

Questions  to  ditto, 73 

Affording  the  History  of  the  Church  in  all  ages,  with  accounts  of  the 
pagan  world  during  Biblical  periods,  and  the  character,  rise,  and  progress 
of  all  Religions,  as  well  as  the  v:irious  sects  of  the  worshipers  of  Christ. 
The  work  is  entirely  non-sectarian,  though  sh-ictly  catholic. 

13 


The  A^utioiial  Sadies  of  Standard  Sc?ioot-7iooks, 

PENMANSH  l"p^ 

1*     ^     0 

Beers'  System  of  Progressive  Penmanship. 

Per  do?eA .$2  50 

This  "round  hand"  system  of  penmanship  in  twelve  nnrabcre  com> 
mends  itself  by  its  simplicity  and  thoroughness.  The  first  fojir  numbers 
are  primary  books.  Nos.  5  to  7,  adyancod  books  for  boys.  Nos.  8  to  10 
advanced  books  for  girls.  Nos.  11  and  12,  ornumenfcil  penmanship. 
These  books  are  printcil  from  steel  plates  (engraved  by  McLccs),  and  are 
unexcelled  fh  mechanical  execution.     Large  quantities  are  aunually  sold. 

Beers'  Slated  Copy  Slips,  per  set *50 

All  beginners  should  practice,  for  a  fww  weeks,  slate  exercises,  familiar* 
izing  them  with  the  form  of  the  letters,  the  motions  of  the  hand  and  arm, 
&c.,  &c.  These  copy  slips,  32  in  number,  supply  all  th«  copies  found  in  a 
complete  series  of  writing-books,  at  a  trifling  cost 

Fulton  &  Eastman's  Copy  Books,  per  dozen  l  50 

A  series  for  the  economical, — complete  in  three  nunihnrs.  (1)  Elemen- 
tary Exercises :    (2)  Gentlemen's  Hand  :     (3)  Ladies' Hand. 

Fulton  k  Eastman's  Chirographic  Charts, 

2  Nos.,  per  set  .     .     . *5  00 

To  enibcllisli  the  school-room  walls,  and  furnish  class  exercise  in  the 
elements  of  Penmanship. 

DRAWING. 


Clark's  Elements  of  Drawing 1  00 

Containing  full  instructions,  with  appropriate  designs  and  copies  for  a 
complete  course  in  this  graceful  art,  from  the  first  rudiments  of  outline  to 
the  finished  sketches  of  landscape  and  scenery. 

Fowle's  Linear  and  Perspective  Drawing      60 

For  the  cultivation  of  the  eye  and  hand,  with  copious  illustrations  and 
directions  which- will  enable  the  unskilled  teacher  to  learn  the  art  himself 
while  instructing  his  pupils. 

Monk's  Drawing  Books— Six  Numbers,  each,    .*    40 

A  series  t)l  proirressive  Drawing  Books,  presenting  copy  and  blank  on 
opposite  pages.  Tlie  copies  are  f  ic-simiics  of  the  best  imported  litho- 
graphs, the  oHginals  of  which  cost  from  50  cents  to  $1..50  each  in  the 
print-stores,  llach  book  contains  eleven  largo  patterns.  No.  1. — Ele- 
mentary stmlics ;  No.  2.— Studies  of  FoliHge;  No.  3. — landscapes;  Na 
4.— Animals,  I. ;  No.  5. — Animals,  IL  ;  No.  U.— Marine  Views,  &c 

Ripley's  Map  Drawing 1  25 

One  of  the  most  efTicient  aids  to  the  acquirement  of  a  knowledge  of 
geography  is  the  practice  of  map  drawing.  It  is  useful  for  the  same  rea- 
son that  th(!  best  exercise  in  orthography  is  the  wnting  of  difficult  words. 
Sight  comes  to  the  aid  of  hearing,  and  a  double  impression  is  produced 
upon  the  memory.  Knowledge  bocomos  less  mechanical  and  more  intui- 
tive. The  student  who  has  sketched  the  outlines  of  a  country,  and  dotted 
the  important  places,  is  little  likely  to  forget  either.  The  impression  pro- 
duced may  be  compared  to  that  of  a  traveler  who  has  been  over  the 
ground — while  more  comprehensive  and  accurate  la  detail 

14 


The  JVati07ial  Series  of  Standard  Schoot-Sooks, 

ELOCUTION. 


Northend's  Little  Orator *60 

Contains  simple  and  attractive  pieces  in  prose  and  poetry,  adapted  to 
the  capacity  of  ciiildren  under  twelve  years  of  age. 

Northend's  National  Orator *l  lo 

About  one  hundred  and  seventy  choice  pieces  happily  arranged.  Tho 
design  of  the  author  in  making  the  selection  has  been  to  cultivate  versa- 
tility  of  expression. 

Northend's  Entertaining  Dialogues .   •   .    .*l  lo 

Extracts  eminently  adapted  to  cultivate  the  dramatic  faculties,  as  well 
as  entertain  an  audience. 

Zachos'  Analytic  Elocution 1  25 

All  departments  of  elocution — such  as  the  analysis  of  the  voice  and  the 
sentence,  phonology,  rhythm,  expression,  gesture,  «&c. — are  hero  arranged 
for  instruction  in  classes,  illustrated  by  copious  examples. 

Sherwood's  Self  Culture l  25 

Self  culture  in  reading,  speaking,  and  conversation — a  very  valuable 
treatise  to  those  who  would  perfect  themselves  in  these  accomplishments. 


BOOK-KEEPING. 


Smith  &  Martin's  Book-keeping l  lo 

Blanks  to  ditto *60 

This  work  is  by  a  practical  teacher  fcnd  a  practical  book-keeper.  It  is 
of  a  thoroughly  popular  class,  and  wCl  be  welcomed  by  every  one  who 
loves  to  see  theory  and  practice  oombined  in  an  easy,  concise,  and 
methodical  form. 

The  Single  Entry  portion  is  well  adapted  to  supply  a  want  felt  in  nearly 
all  other  treatises,  which  seem  to  be  prepared  mainly  for  the  use  of 
wholesale  merchants,  leaving  retailers,  ^  mechanics,  farmers,  &c.,  who 
transact  the  greater  portion  of  the  business  of  the  country,  without  a 
guide.  The  work  is  also  commended  on  this  account  for  general  use  in 
Young  Ladies'  seminaries,  where  a  thorough  grounding  in  the  simpler  form 
of  accounts  will  be  invaluable  to  the  future  housekeepers  of  the  nation. 

The  treatise  on  Double  Entry  Book-keeping  combines  all  the  advan- 
tages of  the  most  recent  methods,  with  the  utmost  simplicity  of  applica- 
tion, thus  affording  the  pupil  all  the  advantages  of  actual  experience  in 
the  counting-house,  and  giving  a  clear  comprehension  of  the  entire  sub- 
ject through  a  judicious  course  of  mercantile  transactions. 

The  shape  of  the  book  is  such  that  the  transactions  can  be  presented  as 
in  actual  practice ;  and  the  simplified  form  of  Blanks,  three  in  number, 
adds  greatly  to  the  ease  experienced  in  acquiring  the  science. 

15 


t.'    i%       ' 


